UNIVERSITY  OF  CALIFORNIA 
LOS  ANGELES 


ELEMENTS 

OF  THE 

KINEMATICS  OF  A  POINT 

AND  THE 

RATIONAL  MECHANICS  OF 
A  PARTICLE 


G.  0.  JAMES,  Ph.D. 

Instructor  in  Mathematics  and  Astronomy 
Washington  University,  St.  Louis 


FIRST  EDITION 

FIRST  THOUSAND 


NEW   YORK 

JOHN   WILEY   &    SONS 

London  :  CHAPMAN  &  HALL,  Limited 

1905 


Copyright,  1905, 

BY 

G.  O.  JAMES 


etmr  OP  MRS.  fran:  •orlet 


ROBERT  DRUMMOND,   PRINTER,  NEW  YORK 


-•-  • 


«••   -•»    • 


Sci&nces 


"""'      J^.3e 


INTRODUCTION 


This  book  is  intended  for  those  who  expect  to  continue 
the  study  of  mechanics  beyond  an  elementary  course,  and 
is  meant  to  serve  as  an  introduction  to  advanced  treatises. 
For  this  reason  special  attention  has  been  given  to  the  prin- 
ciples  and   order   of   presentation,    while   the   applications 
have  been  left  almost  entirely  aside.    No  attempt  has  been 
made  to  avoid  such  mathematical  terms  and  formulae  as 
seemed  necessary,  but  those  problems  requiring  a  knowl- 
J   edge  beyond  the  calculus  and  elementary  differential  equa- 
vj   tions  have  either  been  entirely   omitted,   or  approximate 
^^  solutions  only  have  been  given.     Foucault's  pendulum  has 
been   treated   in    this   way.     Especial   attention   has   been 
^  given  to  relative  motion  and  to  motion  on  the  Earth's  sur- 
^  face,   and  to  obtain  a  proper  orientation   in  the  subject 
>^  the  problems  chosen  have  been  made  as  general  as  possible. 
'  G.  0.  James. 

Washington  University,  May  2,  1904. 


J^  463707 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofkinemaOOjameiala 


CONTENTS 


PART  I 

KINEMATICS 

CHAPTER  I 
THEORY  OF  VECTORS 

ARTICLZ)  PAGE 

1.  Vectors ,. 1 

2.  Equipollence  of  two  vectors  and  notation 2 

3.  Geometric  sum 3 

4.  Analytic  expression  of  geometric  sum 5 

5.  Geometric  difference 8 

6.  Analytic  expression  of  geometric  difference 9 

7.  Decomposition  of  a  vector 9 

8.  Analytic  expression  of  components 10 

9.  Projection  of  a  vector  on  a  line 11 

10.  Analjrtic  expression  of  geometric  sum  referred  to  axes 13 

11.  Geometric  derivatives 14 

12.  Projection  of  the  geometric  derivative  of  a  vector  on  a  plane  and 

on  an  axis 16 

13.  Projection  of  the  geometric  derivative  of  a  vector,  on  the  vector 

itself 19 

14.  Applications 19 

CHAPTER   II 
KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS 

15.  Definitions.     Rest  and  motion 22 

16.  Time 23 

v 


VI  CONTENTS 

ASTICLE  PAGE 

17.  Motion  of  a  point.     Path 23 

18.  Equations  of  motion 23 

19.  Rectilinear  motion 24 

20.  Units  of  time  and  of  length 25 

21.  Case  where  it  is  necessary  to  specify  the  units 25 

22.  Homogeneous  equations 26 

23.  Change  of  units 26 

24.  Uniform  rectilinear  motion.     Velocity 28 

25.  Equation  of  uniform  motion 29 

26.  Accelerated  retilinear  motion 30 

27.  Numerical  examples 30 

28.  Knots  and  nautical  miles 31 

29.  Equation  of  accelerated  motion  in  terms  of  the  mean  velocity.  .  .  32 

30.  Acceleration 32 

31    Equation  of  uniformly  accelerated  motion 33 

32.  Discussion  of  the  motion  by  means  of  the  equation  of  motion  34 

33.  Case  where  the  displacement  is  taken  as  the  independent  variable  35 

CHAPTER  III 
APPLICATIONS  TO  ORDINARY  RECTILINEAR  MOTIONS 

A.  Uniform  Motion 

34.  Properties  of  uniform  motion 37 

35.  Problem 38 

B.  Uniformly  Accelerated  Motion 

36.  Properties  of  uniformly  accelerated  motion 39 

37.  Remark 42 

38.  Equations  obtained  by  taking  the  space  described  as  independent 

variable 42 

C.  Periodic  Motions 

39.  Equations  of  periodic  motion 43 

40.  Equations  of  harmonic  motion 44 

41.  Amplitude,  elongation,  frequency,  argument,  phase 45 

42.  Velocity  and  acceleration 47 

43.  Discussion  of  the  motion 47 

44.  Influence  of  the  phase 49 


CONTENTS  VU 


CHAPTER  IV 
RELATIVE  MOTION  ALONG  A  LINE 

ARTICLE  PAGE 

45.  Relative  motion : 51 

46.  Equation  of  relative  motion 52 

47.  Relative  velocity  and  acceleration 52 

48.  Apparent  motion 53 

49.  Convective  motion 54 

50.  Simultaneous  motions 55 

51.  Composition  of  harmonic  motions  of  same  period 56 

52.  Representation  of  the  amplitude  and  phase  by  a  vector 57 

53.  Equation  of  the  resultant  motion 58 

54.  Vibrations  of  different  periods 59 


CHAPTER  V 
VELOCITY   AND    ACCELERATION    IN    CURVILINEAR   MOTION 

A.  Velocity 

55.  Displacement 61 

56.  Velocity 62 

57.  Algebraic  value  of  the  velocity 63 

58.  Projection  of  the  velocity  on  the  displacement 64 

59.  Geometric  representation  of  the  path 65 

60.  Projection  of  the  velocity  on  any  axis 65 

B.  Acceleration 

61.  Acceleration.     Hodograph 66 

62.  Projection  of  the  acceleration  on  any  axis 67 

63.  Projection  of  the  acceleration  on  the  tangent  and  principal  nor- 

mal    67 

64.  Projection  of  the  acceleration  on  the  binormal 68 

65.  Condition  that  the  acceleration  be  constantly  tangential  or  nor- 

mal to  the  path.      Condition  that  the  acceleration  be  con- 
stantly zero 69 

66.  Composition  of  motions  along  the  same  path 69 


Vm  CONTENTS 


CHAPTER  VI 

ANGULAR  AND  AREAL  MOTION.      EQUATIONS  AND  GENERAL 

THEOREMS 

ARTICLE  PAOB 

67.  Angular  displacement 70 

68.  Angular  velocity   71 

69.  Relation  between  linear  and  angular  velocities 72 

70.  Angular  acceleration 72 

71.  Relation  between  linear  and  angiilar  accelerations 73 

72.  Areal  velocity 74 

73.  Extension  of  theorems  already  found 76 

74.  Equations  of  motion.     Units  of  time  and  displacement.     Homo- 

geneity   76 

75.  Equation  of  uniform  motion 77 

76.  Equation  of  uniformly  accelerated  motion 78 

77.  Equation  of  motion  when  angular  acceleration  is  not  constant  78 

78.  Properties  of  uniformly  accelerated  angular  motion 78 

79.  Periodic  angular  motion.     Harmonic  motion 79 

80.  Relative  angular  motion 80 

81.  Composition  of  harmonic  angular  motions  of  same  period.  ....  82 


CHAPTER  VII 
MOTION  REFERRED  TO  COORDINATE  AXES 

82.  Important  remark 83 

83.  Projection  of  the  motion  on  a  plane  and  on  an  axis 83 

84.  Projection  of  the  velocity  and  acceleration 84 

85.  Equations  of  motion  referred  to  the  coordinate  axes 85 

86.  Equations  of  the  projected  motion  on  the  coordinate  planes 86 

87.  Projection  of  the  path 86 

88.  Equations  of  the  projected  motion  on  the  coordinate  axes 87 

89.  Projections  of  the  velocity 87 

90.  Equations  of  motion  of  the  point  which  describes  the  hodograph  88 

91.  Projections  of  the  acceleration  on  the  coordinate  axes 88 

92.  Resume 89 

93.  Most  general  motion  in  which  the  projected  motions  are  uniformly 

accelerated 90 

94.  Motion  in  which  the  projected  motions  are  harmonic 91 

95.  Rectilinear  harmonic  motion  considered  as  the  projection  of  uni- 

fonn  circular  motion 93 


CONTENTS  ix 


CHAPTER   VIII 
RELATIVE  MOTION.     MOVING  AXES 

ARTICLE  PAOB 

96.  Fixed  and  moving  axes 95 

97.  Equations  of  absolute  and  relative  mdtion 96 

98.  Statement  of  the  problem 96 

99.  Motion  of  the  relative  axes 96 

100.  Solution  of  the  problem 97 


PART  II 

MECHANICS 

CHAPTER  IX 
MECHANICS  OF  A  FREE  PARTICLE 

101.  Material  point  or  particle 101 

102.  Purpose  of  this  book 102 

103.  Problem  of  mechanics 103 

104.  The  role  of  observation  and  experiment 104 

105.  Principles  of  mechanics 104 

106.  The  absolute  axes.     Isolated  particle 104 

107.  First  principle  of  mechanics 105 

108.  Meaning  of  the  first  principle 105 

109.  Field  of  force 105 

110.  Uniform  field  of  force 106 

111.  Constant  field  of  force ' 106 

112.  Superposition  of  fields  of  force 106 

113.  Second  principle  of  mechanics 106 

114.  Meaning  of  the  second  principle 106 

115.  Reaction  of  the  particle  on  the  field 107 

116.  Third  principle  of  mechanics 107 

117.  Meaning  of  third  principle 108 

118.  Properties  of  the  coefficients  p^^- 108 

1 19.  Definition  of  mnss 109 

120.  Force 110 

1 21 .  Observations  on  the  notion  of  force 110 

1 22.  Composition  of  forces.     Resultant Ill 

123.  Decomposition  of  forces.     Components 112 

124.  Equilibrium  of  a  free  particle 113 


X  CONTENTS 

ARTICLE  PAGE 

125.  Force  acting  on  a  particle  at  rest 114 

126.  New  definition  of  direction  of  force *. 114 

127.  Tangential  and  normal  force 115 

128.  "Resistance 115 

CHAPTER  X 
THE  UNITS  OF  MECHANICS 

129.  The  units  of  kinematics 116 

130.  The  units  of  mechanics 116 

131.  The  three  fundamental  units.     Derjved  units 116 

132.  Systems  of  units 117 

133.  cIg.S.  system 117 

134.  M.K.S.  system 117 

135.  English  system 117 

1?6.  Units  of  astronomy 117 

137.  Unit  of  force 117 

13^.  Remark 118 

CHAPTER   Xr 
GENERAL  NOTIONS  ON  THE  FORCES  OF  NATURE 

139.  Principle  of  the  equality  of  action  and  reaction 119 

140.  Analytic  expression  of  the  common  force  between  two  free  par- 

ticles   120 

141.  Central  forces 121 

142.  Universal  gravitation 121 

143.  Note 122 

CHAPTER   XII 

DETERMINATION  OF  THE  LAW  OF  FORCE  FROM  THE 
MOTION   PRODUCED 

144.  Finite  and  dififerential  equations  of  motion 123 

145.  The  two  general  problems  of  the  mechanics  of  a  particle  in  terms 

of  force 124 

146.  The  first  problem 125 

147.  The  different  forces  capable  of  producing  a  given  rectilinear 

motion 1 26 

148.  The  force  producing  uniformly  accelerated  motion 127 


CONTENTS  XI 

ARTICLE  PAOB 

149.  Law  of  force  porduclng  harmonic  motion 127 

150.  Law  of  force  producing  harmonic  motion  with  coefficient  of 

extinction 1 27 

151.  On  the  resistance  producing  extinction 128 

152.  Law  of  force  producing  uniform  curvilinear  motion 129 

CHAPTER  XIII 
MOTION  PRODUCED  BY  FORCES  OBEYING  KNOWN  LAWS 

153.  Statement  of  the  problem 131 

154.  Attraction  toward  a  fixed  centre  proportional  to  the  distance  133 

CHAPTER  XIV 

RELATIVE  MOTION 

155.  Relative  axes 141 

156.  Relative  force 141 

157.  Problems  of  relative  motion 141 

158.  First  problem 142 

159.  Second  problem 143 

CHAPTER   XV 
MOTION  OF  A  FREE  PARTICLE  ON  EARTH'S  SURFACE 

160.  The  relative  axes 145 

161.  Differential  equations  of  motion 146 

162.  First  approximation 146 

163.  Second  approximation 147 

CHAPTER  XVI 

CONSTRAINED  MOTION  OF  A  PARTICLE.     GENERAL 
CONSIDERATIONS 

164.  Constrained  particle 1 49 

165.  Mathematical  expression  of  the  conditions  of  constraint 149 

H6.  Variable  constraints 1 50 

1 67.  Smooth  constraints 150 

16^.  Forces  of  constraint 1 50 

169.  Motion  of  a  constrained  particle 151 


xii  CONTENTS 

ARTICLE  PAGE 

170.  Direction  of  Fr 151 

171.  Magnitude  of  Fr 1 52 

172    Particle  at  rest  on  a  smooth  surface 152 

173.  Curve  of  constraint 152 

174.  Particle  at  rest  on  a  smooth  curve 153 

175.  Conditions  of  equilibrium 153 

CHAPTER  XVII 
CONSTRAINED  MOTION  ON  EARTH'S  SURFACE 

176.  Variation  of  g  with  latitude 154 

177.  Particle  falling  on  smooth  curve 1 56 

17S.  Oscillations  of  a  simple  pendulum 1 59 

1 70.  Oscillations  of  small  amplitude 161 

180.  Velocity  in  oscillations  of  any  amplitude 162 

181.  Tension  in  the  string 162 

182.  Foucault's  pendulum 163 

183.  Approximate  solution.     Particular  case.     General  case.     Fou- 

cault's case.    Theorem  of  Chevilliet 166-1 70 


ELEMENTS    OF    KINEMATICS    AND 
MECHANICS 


CHAPTER   I 
THEORY  OF  VECTORS 

1.  Vectors. — 2.  EquipoUence  of  two  vectors  and  notation. — 3.  Geo- 
metric sum. — 4.  Analytic  expression  of  geometric  sum. — 5.  Geo- 
metric difference. — 6.  Analytic  expression  of  geometric  differ- 
ence.— 7.  Decomposition  of  a  vector. — 8.  Analytic  expression 
of   the   components. — 9.  Projection   of   a   vector   on   a   line. — 

10.  Analytic  expression  of  geometric  sum  referred  to  axes. — 

11.  Geometric  derivatives. — 12.  Projection  of  the  geometric 
derivative  of  a  vector  on  a  plane  and  on  an  axis. — 13.  Projection 
of  the  geometric  derivative  of  a  vector  on  the  vector  itself. — 
14.  Applications. 

I.  Vectors. — Many  of  the  quantities  of  mechanics  can 
be   conveniently   represented   by  seg- 
ments of  straight  lines  having  a  defi- 
nite  direction    and    length.     Such    a 
segment  is  termed  a  vector. 

A  vector  is  a  segment  AP  (fig.  1)  of 
a  straight  Une  having  an  origin  A  and 
an  extremity  P.    It  is  defined  by  the  Fig.  1. 

following  elements: 


2  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

1°  Its  origin  A; 

2°  Its  length  AP; 

3°  Its  direction,  which  is  by  definition  the  direction  of 
the  hne  AP  from  A  toward  P. 

The  vector  is  always  written  by  placing  the  letter  repre- 
senting the  origin  first.  Thus  the  vector  AP  has  its  direc- 
tion from  A  to  P,  while  the  vector  PA  has  its  direction  frc»m 
P  toward  A. 

This  is  expressed  by  writing 

PA=-AP  [1] 

Ordinarily,  a  vector  AP  is  denoted  by  the  single  letter  P 
and  an  arrow  placed  at  P  to  denote  the  direction. 

2.  EquipoUence  of  two  vectors  and  notation. — Two  vec- 
tors equal,  parallel,  and  having  the  same  direction  are  termed 
equipollent.  This  geometric  equivalence  is  expressed  by 
writing  a  dash  over  the  two  vectors,  as 

Pl=P~2  [2] 

which  asserts  that  Pi  is  equipollent  to  Pa-  Two  ■^'ectors 
which  are  equal  and  parallel  but  have  opposite  directions 
are  termed  equal  and  opposite,  and  this  is  expressed  by  the 
equipollence 

Pi  =  -P2  [3] 

If  they  have  the  same  origin,  or  are  situated  on  the  same 
straight  line,  they  are  termed  equal  and  directly  opposite. 
Figure  2  illustrates  the  different  cases  of  equal  and  directly 
opposite  vectors.    This  is  again  expressed  by  the  equipollence 

F^=--F2  [4] 


THEORY  OF  VECTORS  3 

which,  it  will  be  observed,  takes  no  account  of  the  origins 
of  the  vectors. 


Fig.  2. 

3.  Geometric  sum,  or   composition  of  vectors. — Consider 
(fig.  3i)  a  system  of  vectors 

^1,  P2,  •  •  •  Pn-l,  Pn 

distributed  in  any  manner  in  space.  Displace  them  parallel 
to  themselves  so  as  to  place  them  end  to  end  in  any  order — 
in  the  order  of  then*  indices,  for  instance.  To  fix  the  ideas 
suppose  n  equal  to  six.  By  this  means  there  is  formed  an 
open  polygonal  contour  (fig.  82),  plane  or  twisted,  of  which 
the  sides 

1,  2,  3,  4,  5,  6 

are  equipollent  to  the  given  vectors.  This  polygon  is  termed 
the  polygon  of  the  given  vectors,  and  the  vector  APq,  which 
closes  it,  is,  by  definition,  the  geometric  sum  of  the  given 
vectors.  Call  this  sum  S.  Then  this  definition  is  expressed 
analytically  by  the  equipollence 


S=Pi+P2+P3+P4  +  P5  +  P6 


4  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

and  in  the  general  case  by 

i  —  n 

1  =  1 


[5] 


Remark. — ^This  extension  of  the  word  sum,  extremely 
convenient  in  mechanics,  is  justified  l)y  the  fact  that  these 
geometric  sums  possess  the  same  properties  as  arithmetic 


and  algebraic  sums,  which  they  include  as  particular  cases. 
This  results  from  the  following  geometric  theorems  which 
are  easily  proven,  but  which  I  shall  not  stop  to  verify. 

Theorem  I. — The  geometric  sum  of  a  system  of  rectors 
is  independent  of  the  order  of  summation. 

Theorem  II. — The  geometric  sum  of  a  system  of  vectors 
is  not  changed  by  replacing  certain  of  them  by  their  partial 
sums,  or  conversely  by  replacing  certain  of  them  by  others  of 
which  they  are  the  partial  sums. 

Theorem  IIIi. — //  all  the  vectors  of  a  system  be  increased 
or  decreased  in  the  same  ratio,  their  geometric  sum  is  increased 
or  decreased  in  this  ratio. 


THEORY  OF  VECTORS  5 

Theorem  III2. — To  multiply  the  geometric  sum  hy  any 
number,  it  is  sufficient  to  multiply  each  vector  of  the  system 
by  this  number. 

4.  Analytic  expression  of  geometric  sum. — 1°  Two  vec- 
tors.— ^Forming   the   polygon    (fig.   4)    of   the   two   vectors, 


Fig.  4. 


Pi  and  P2,  this  reduces  to  a  triangle.  I^et  a  be  the  angle 
between  the  positive  directions  of  Pi  and  P2.  Their  geomet- 
ric sum  is,  by  definition,  the  third  side  of  this  triangle,  and 

The  magnitude  of  S  is  given  by  the  algebraic  equation 
52  =p^2  +p^2^2PiP2  cos  a 

To  determine  6,  the  angle  between  the  positive  directions 
of  S  and  Pi,  we  have  the  relation 

P2  _    S 
sin  d    sin  a. 


6  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

p 
Whence  sin  ^  =  -^  sin  a  [6] 

Also  P-J^ ^Pi^+S^-2PiS  cos  d 

S2+p^2_p.J2 


Whence  cos  6  ■■ 


2PiS 


a      Pi+P2C'0Sa  r^, 

or  cos  6  = [7] 


Since  0°  ^  ^  ^  180°,  [7]  determines  which  quadrant  it  lies  in. 

Remark. — From  the  figure  it  is  evident  that  if  we  com- 
plete the  parallelogram  having  Pi  and  P2  as  sides,  the  vector 
P2  is  equipollent  to  P2,  and  S  is  the  diagonal  of  the  paral- 
lelogram having  Pi  and  P2'  for  sides,  whence  the 

Theorem. — //  from  the  same  point  two  vectors  he  drawn 
equipollent  to  the  given  vectors,  the  diagonal  of  the  parallelo- 
gram constructed  on  these  two  vectors  as  sides  is  equipollent 
to  the  geometric  sum  of  the  given  vectors. 

Corollary. — If  a  =90°,  then 

52=Pi2+P22 


and  cos^==-^ 

o 


p 
or  tan^=^ 

Pi 

2°  Three  vectors. — The  geometric  sum  S  (fig.  5)  of  three 
vectors  Pi,  P2,  and  P3  is,  by  definition,  the  vector  AS. 
Denoting  by  a,  /?,  ^  the  angles  between  the  positive  direc- 


THEORY  OF  VECTORS 


tions  of  Pi  and  P-z,  P2  and  P3,  P3  and  Pi,  the  magnitude 
of  S  is  given  by 

^2  =  p^2  +  p^2  +  P32  +  2P1P2  cos  a 

+ 2P2P3  cos  /? + 2P3P1  COS  r    [8] 


Fig.  5. 


The  angles  di,  62,  Os  made  by  the  positive  direction  of  S 
and  the  positive  directions  of  Pi,  P2,  and  P3  are  deter- 
mined by  the  relations 


Pl^+S2-2PiS  cos  di  =P22+P32+2P2P3  COS  /? 

P22  +  S2-  2P2S  cos  O2  =  P32 + Pi2  +  2P3P1  cos  r  ■ 
Ps^  +  S2-  2P3S  cos  O3  =  Pi2  +  P22  +  2P1P2  cos  a  J 


[9] 


Remark. — It  is  again  evident  from  the  figure  that  if  at 
at  A  vectors  P2'  and  P3'  be  drawn  equipollent  to  P2  and 
P3,  the  vector  AS  is  the  diagonal  of  the  parallelopiped 
constructed  on  ^Pi,  ^P2,  AP3  as  edges,  whence  the 

Theorem. — //  from  the  same  point  vectors  be  drawn  equi- 
pollent to  three  given  vectors,  the  diagonal  of  the  parallelopiped 
constritcted  on  these  as  edges  is  equipollent  to  the  geometric 
sum  of  the  given  vectors. 


8  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

5.  Geometric  difference. — ^The  ^geometric  difference  of  two 
veccors  Pi  and  P2  is,  hy  definition,  the  vector  which  added 
to  one  of  them  gives  the  other. 

Subtracting  Pi  jrom  P2  and  calhng  the  difference  Q, 
we  have  the  equipoUence 

From  Art.  4  it  follows  that  P2  is  equipollent  to  the  diag- 
onal of  the  parallelogram  having  Pi  and  Q  as  sides.  If 
then  from  A  (fig.  6),  the  origin  of  Pi  and  P2,  a  vector  —Pi, 


Fig.  6. 

equal  and  opposite  to  Pi,  be  drawn,  Q  is  equipollent  to 
the  diagonal  of  the  parallelogram  having  —Pi  and  P2  for 
sides.  Q  is  then  the  geometric  sum  of  P2  and  —Pi,  which 
is  expressed  by  the  equipoUence 

^=P2  +  (-Pr) 

or  Q=P2-Pi  [10] 

which  may  be  looked  upon  as  a  consequence  of  the  equipol- 
lence 

P2=Pi+e 

It  follows,  then,  that  an  equipoUence  involving  three  vectors 
may  be  treated  as  an  algebraic  equation  so  far  as  the  trans- 


THEORY  OF  VECTORS  9 

position  of  its  terms  is  concerned,  and  by  similar  reason- 
ing an  equipollence  involving  any  number  of  vectors  may  be 
so  treated. 

It  is  evident  from  the  figure  that  the  vector  P1P2  is 
also  equipollent  to  Q,  and  we  have  then  the  following  geo- 
metrical construction  for   the  difference  of  two  vectors. 

From  a  point  A  draw  two  vectors  Pi  and  P2  equipollent 
to  the  given  vectors  and  complete  the  triangle  by  connecting 
their  extremities.  The  third  side  of  this  triangle  is  equipol- 
lent to  the  difference  of  the  two  vectors.  The  direction  of  this 
difference  is  from  Pi  to  P2  if  Pi  is  subtracted  from  P2,  and 
from  P2  to  Pi  if  P2  is  subtracted  from  Pi. 

6.  Analytic  expression  of  geometric  difference. — If  a  be 
the  angle  between  Pi  and  P2,  then 

Q2=p,2  +  p^2_2PiP2  cos  a  [11] 

which  determines  the  arithmetic  value  of  Q. 

Replacing  Pi  by   —Pi  and  5  by  Q  in  [7],  we  have 

a       — P1+P2COS0:  riDi 

cos  d  =- . — [12] 

which  determines  the  angle  0  between  Pi   and  Q. 

7.  Decomposition  of  a  vector. — To  decompose  a  vector 
S  is,  by  definition,  to  find  a  system  of  vectors 

Pi  . . .  P« 

of  which  S  is  the  geometric  sum. 

A  system  of  two  vectors  Pi  and  P2  of  which  S  is  the 
geometric  sum  is  easily  found.  In  fact,  it  is  sufficient  to 
construct  any  parallelogram  of  which  S  is  the  diagonal. 


10 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


The  sides  Pi  and  P2  of  this  parallelogram  are  then  equi- 
pollent to  the  required  vectors.  Similarly  to  decompose 
a  vector  S  into  a  system  of  three  vectors  it  is  sufficient  to 
construct  a  parallelopiped  of  which  S  is  the  diagonal.  The 
three  edges  of  this  parallelopiped  issuing  from  the  vertex 
which  is  the  origin  of  S  are  then  equipollent  to  the 
required  vectors.  Since  an  infinite  number  of  such  paral- 
lelograms or  parallelopipeds  can  be  constructed,  the  decom- 
position can  be  effected  in  an  infinite  number  of  ways,  and 
to  obtain  a  unique  decomposition  one  vector  must  be  assigned 
in  the  parallelogram  and  two  in  the  parallelopiped. 

Instead  of  assigning  one  or  two  of  the  vectors,  two  or 
three  of  the  directions  may  be  assigned,  and  the  decom- 
position is  unique  since  there  can  be  but  one  parallelogram 
or  parallelopiped  with  assigned  directions  for  its  edges. 

8.  Analytic  expression  of  the  components. — 1°  If  one  of 
the  component  vectors.  Pi  say,  is  assigned  in  magnitude 

and  direction,  the  magnitude  and 
direction  of  P2  (fig.  7)  are  given 
by 

P^2  =  pj2  +  S^  -  2PiS  COS  ^1   1 


sin  ^2=  sin  ^1-^ 


[13] 


2°  If  the  two  directions  di 
and  62  are  assigned,  then  the 
magnitudes  are  given  by  the  equa- 
tions 


-^^       sin  (^1+^2) 


P2=S  . 


sm  Oi 


'sin  (61+62) 


[14] 


THEORY  OF  VECTORS 


11 


Remark.  —  The  analytic  expressions  for  the  direc- 
tions and  magnitudes  when  S  is  decomposed  into  three 
components  Pi,  P2,  and  P3  are  easily  obtained  from  the 
geometry  of  the  parallelopiped. 

If  ^1+^2=90°,  or  the  assigned  directions  are  orthogonal, 
then 

Pi  =S  cos  6 1 

P2=Scosd2 

9.  Projection  of  a  vector  on  a  line. — ^The  projection  of  a 
vector  AP  on  a  line  OX  (fig.  8)  is  the  vector  AxPx  obtained 
by  projecting  the  line  AP  on  OX.  The  projection  of  A  is, 
by  definition,  the  origin  of  AxPx' 


Fig.  8. 


Denoting  by  6  the   angle  between  P  and  OX,  we  have 

Px  =  P  cos  6  [151 

Theorem  I. — The  algebraic  sum  oj  the  'projections  of  two 
vectors  on  a  line  is  eqtud  to  the  projection  0}  the  geometric 
sum  of  the  two  vectors  on  that  line. 


12  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Let  (fig.  9)  Pi  and  P2  be  the  given  vectors  and  S  their 
geometric  sum.  Then  A^Px,,  A^Px,,  and  AxSx  are  their 
projections  on  OX, 


and 


A  xSx  —  A  xPxt  +  PxiSx 


Fig.  9. 


But  PiS  is  equipollent  to  AP2,  and  therefore 


P  xJ^x  —  AxPxi 


Hence 


AxSx    —AxPx,^AxPa 


or  denoting  by  6,  61,  and  62  the  angles  between  S,  Pi,  P2, 
and  OX 


S  cos  0  =  Pi  cos  ^1  +  P2  cos  62 


[16] 


This  same  reasoning  can  be  extended  to  a  system  of 
n  vectors  by  first  taking  two  of  them,  then  their  geometric 


THEORY  OF  VECTORS  13 

sum  and  a  third  and  so  on  until  the  last  vector  is  reached, 
whence 

Theorem  II. — The  algebraic  sum  of  the  projections  of  a 
system  of  n  vectors  on  a  line  equals  the  projection  of  the 
geometric  sum  of  the  system  on  that  line,  or 

S  cos  ^  =Pi  cos  di+   ...    +Pn  cos  dn 


or  S  cos  e^^Pi  cos  di  [17] 

1 


10.  Analytic  expression  of  geometric  sum  referred  to 
axes. — The  two  theorems  of  the  last  section  give  a  simple 
method  of  determining  the  geometric  sum  of  a  system  of 
vectors.  Choosing  the  three  perpendicular  axes  OX,  OY, 
and  OZ  and  projecting  all  the  vectors  of  the  system  on 
these  axes,  the  sum  of  the  projections  on  each  axis  gives 
the  projection  of  the  geometric  sum  on  that  axis,  which 
is  the  edge  of  the  rectangular  parallelopiped  of  which  the 
geometric  sum  is  the  diagonal  and  since  when  the  three 
edges  are  determined  the  diagonal  is  determined,  the  geo- 
metric sum  is  determined  in  magnitude  and  direction. 

Writing  >Sx  =  i'Px 

S^  =  IP^ 
then  S^=SJ'+Sy^+SJ^  '         [18] 

and  calling  Qx,  Oy,  and  0^  tlie  angles  between  S  and  OX, 
OY,  OZ 


14 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


COS  Or  =  -, 


5.1 
S 


Sy 

"  S 

a        ^1 
COS  02=-^ 


COS     Oy 


[19] 


Corollary. — If  the  vectors  all  lie  in  one  plane,  then 
referring  them  to  two  rectangular  axes  OX  and  OF  in  that 
plane 


COS  6x  = 


S 


COS  dy  =  -^ 


II.  Geometric  derivatives.  —  1°  Vector  with  a  fixed 
origin. — Consider  AP  (fig.  10)  whose  origin  A  is  fixed  and 
whose  extremity  P  describes  a  curve  C. 

Let  AP  be  its  position  at  the  instant  t  and  APi  its  posi- 
tion at  the  instant  t  +  Jt. 

The  geometric  increment  of  P  during  the  time  Jt  is  the 
geometric  difference 


PPi=Pi-P^JP 


The  vector 


PE^ 


JP 

At 


THEORY  OF  VECTORS 


15 


tends  to  the  limiting  vector  PP' ,  tangent  to  C  at  P,  as  M 
tends  to  zero,  so  that 

and  this  vector  PP'  is  called  the  geometric  derivative  of  P 
at  the  instant  t. 


Fig.  10. 


I  shall  use  the  svmbol 


D 
Dl 


to  denote  a  geometric  derivative  in  distinction  from  the 
symbol 

A 

dt 
used  for  algebraic  derivatives. 


16  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

We  may  then  write 


DP    ,.       AP' 


[20] 


2°  Vector  with  moving  origin. — To  obtain  the  geometric 
derivative  of  the  vector  PP'  with  moving  origin  P  (fig.  11) 

at  the  instant  t  we  construct 
at  the  fixed  point  P  a  vector 
equipollent  to  the  vector  PP' 
and  take  its  geometric  deriva- 
tive, and  this  is  defined  as 
the  second  geometric  deriva- 
P/    tive  of  AP,  so  that 


Fig.  11. 


DP'    D^P 

pp"=Th-W      [21] 


where  PP"  represents  a  vector  equipollent  to  the  geometric 
derivative  of  PP'. 

12.  Projection  of  the  geometric  derivative  of  a  vector 
on  a  plane  and  on  an  axis.  —  Let  the  moving  vector  AP 
(fig.  12)  be  projected  on  the  fixed  plane  II,  and  denote  pro- 
jections on  this  plane  by  the  symbol  (  )„ 


Then 


or 


(PPO^=pp, 


Hence 


/JP\  _Jp 
\At).~At 


THEORY  OF  VECTORS 


17 


since  AP  and  —rr  are  vectors  making  the  same  angle  with  U. 


/JP' 
Therefore  lim  [-jti 


=lim 


IT 


'  J<=0 


or 


DP\      Dp 

Dt  I  rot 


122] 


FiQ.  12. 


Hence,  the  projection  on  any  plane  of  the  geometric  deriv- 
ative of  a  vector  is  equipollent  to  the  geometric  derivative  of 
the  projection  of  the  vector  on  the  plane. 

Let  OX  be  any  axis — not  necessarily  connected  with 
the  plane  77,  but  for  convenience  taken  in  that  plane.  Denote 
projections  on  OX  by  the  symbol  (  )x. 


18  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Then  (PPijx=Xi~-x 

or  (JP)^  =  Jx 


Hence  I  -rr  I   = 


Jt/x     At 


and  lim  (  -77  I  =lini  — - 

.    '  \AtJxUt=o  Jtjjt^o 


/DP\       dx    d{x-a') 


Hence,  the  projection  of  the  geometric  derivative  of  a 
vector  on  an  axis  equals  the  algebraic  derivative  of  the  pro- 
jection of  the  vector  on  that  axis. 

If  d  denote  the  angle  that  AP  makes  with  OX  then 

x  —  a'=r  cos  d 


.  d{x—a)     dr        ^         .      dd 

and  — 1: =  37  cos  d  —  r  sni  d^- 

dt  dt  dt 


where  r  is  the  length  of  AP. 

/DP\      dr        ^        .      dd 
^""'^  \-Dt)rdt'''''^-'''''^di.  ^24] 


THEORY  OF  VECTORS  19 

Corollary.  —  If    AP  be  perpendicular    to   OX,   then 
^=90°  and 


Qr-(^) 


dt     e  = 


fl=90° 


13.  Projection  of  the  geometric  derivative  of  a  vector 
on  the  vector  itself. — Making  (9  =  0  in  [24]  we  have  for  the 
projection  of  the  geometric  derivative  of  a  vector  on  the 
vector  the  expression 


(f)„4:         f-i 


14.  Applications. — First  'problem:  What  is  the  necessary 
and  sufficient  condition  that  the  geometric  derivative  of  a  vec- 
tor he  constantly  directed  along  the  vector? 

In  this  case  the  geometric  derivative  equals  its  projec- 
tion on  the  vector  itself, 


DP     (DP\      dr 
whence  Dt=KDt),^dI 


Hence,  projecting  on  the  X-axis, 

dr 


\Dt  /, 


J,  cos^ 
dt 


and  therefore 


dr  dd    dr 

37  cos  ^  — r  sm  d-jr  =  ttCOS  0 

dt  dt     dt 


20  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

whence  rsinPTr=0 

at 


or  0  =  do,  a  const. 

Therefore,  the  necessary  and  sufficient  condition  that 
the  geometric  derivative  of  a  vector  he  constantly  directed  along 
the  vector  is  that  the  orientation  of  the  vector  be  constant. 

Second  problem:  What  is  the  necessary  and  sufficient 
condition  that  the  geometric  derivative  of  a  vector  be  constantly 
normal  to  the  vector? 

In  this  case  the  projection  of  the  geometric  derivative 
on  the  vector  is  zero, 


dr 
whence  ,,  =0 

dt 


and  T=ro,  a  const. 

Therefore,  the  necessary  and  sufficient  condition  that 
the  geometric  derivative  of  a  vector  be  constantly  normal  to 
the  vector  is  that  the  length  of  the  vector  be  constant. 

Third  problem:  What  is  the  necessary  and  sufficient  con- 
dition that  the  geometric  derivative  of  a  vector  be  zero  ? 

If  the  geometric  derivative  be  zero,  then  its  projection 
on  any  axis  is  zero, 


dx    ^ 
whence  37  =  0 

(U 


or  x=Xo,  a  const. 


THEORY  OF  VECTORS  21 

similarly  t/=?/o,  a  const. 

z=ZOf  a  const. 

and  hence  the  extremity  P  is  fixed. 

Therefore,  the  necessary  and  sufficient  condition  that  the 
geometric  derivative  of  a  vector  be  constantly  zero  is  that  the 
vector  be  constant  both  in  orientation  and  magnitude. 


CHAPTER  II 

KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS 

15.  Definitions.  Rest  and  motion. — 16.  Time. — 17.  Motion  of  a 
point.  Path.  —  18.  Equations  of  motion.  —  19.  Rectilinear 
motion. — 20.  Units  of  time  and  of  length. — 21.  Case  where  it  is 
necessary  to  specify  the  units. — 22.  Homogeneous  equations. — 
23.  Change  of  units. — 24.  Uniform  rectilinear  motion.  Velocity. — 
25.  Equation  of  uniform,  motion. — 26.  Accelerated  rectiHnear 
motion. — 27.  Numerical  examples. — 28.  Knots  and  nautical 
miles.—  29.  Equation  of  accelerated  motion  in  terms  of  the 
mean  velocity. — 30.  Acceleration. — 31.  Equation  of  uniformly 
accelerated  motion. — 32.  Discussion  of  the  motion  by  means 
of  the  equation  of  motion. — 33.  Case  where  the  displacement 
is  taken  as  independent  variable. 

15.  Definitions.  Rest  and  motion. — When  a  point  is 
spoken  of  as  at  rest  or  in  motion  it  is  always  understood 
that  this  rest  or  motion  takes  place  with  respect  to  other 
points  in  its  neighborhood.  A  point  immovable  on  the 
Earth's  surface  is  at  rest  with  respect  to  all  other  fixed 
points  on  the  Earth,  but  is  moving  with  respect  to  points 
on  the  Sun,  since  the  Earth  is  itself  moving  relative  to  the 
Sun.  Hence,  any  relative  motion  is  observable,  but  it  is 
convenient  in  kinematics  to  choose  some  system  of  points 
to  which  the  motion  of  all  other  points  may  be  referred 
to  regard  this  system  as  absolutely  fixed,  by  definition. 
Motion  with  respect  to  this  system  is  then  termed  absolute 
motion.    Thus,  for  points  on  the  Earth,  a  system  of  points 

22 


KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS.       23 

immovable  on  the  Earth's  surface  may  be  chosen  as  the 
absolutely  fixed  system,  and  the  motion  of  all  other  points 
referred  to  them.  To  discuss  the  motion  of  the  Earth  and 
planets  a  second  system  of  absolutely  fixed  points  must  be 
chosen.  These  may  be  conveniently  taken  on  the  Sun, 
while  to  discuss  the  motion  of  the  solar  system  the  so-called 
fixed  stars  must  be  chosen  as  points  of  reference.  There 
being  no  system  of  points  to  which  the  motion  of  these 
stars  can  be  referred,  discussion  cannot  go  beyond  them. 

But  while,  in  kinematics,  the  choice  of  the  absolutely 
fixed  system  is  perfectly  arbitrary,  it  is  no  longer  so  in 
mechanics,  and  there  we  shall  see  that  the  fixed  stars  mu^t 
be  chosen  as  the  system  of  reference. 

i6.  Time. — To  define  the  time  when  the  moving  point 
is  at  a  given  position,  it  is  referred  to  a  certain  instant, 
called  the  initial  instant,  and  the  number  of  mean  solar 
seconds  before  or  after  the  initial  instant  is  given.  Hence 
a  time  t  designates  an  instant  t  seconds  before  or  t  seconds 
after  the  initial  instant  according  as  t  is  positive  or  negative, 
that  is,  according  as  it  has  the  plus  or  minus  sign  prefixed. 

17.  Motion  of  a  point.  Path. — A  point  is  said  to  move 
with  respect  to  some  arbitrarily  fixed  system  of  points 
when  at  least  one  of  the  distances  between  it  and  the  points 
of  the  fixed  system  changes.  The  curve  described  by  the 
moving  point  is  called  its  path  or  trajectory. 

18.  Equations  of  motion. — Referring  the  point  to  a  sys- 
tem of  rectangular  axes  0 — XYZ,  its  coordinates  will  vary 
with  the  time  and  are  therefore  functions  of  t.  Suppose 
them  to  be  given  by  the  following  equations: 

x  =  m       y  =  x(t)       z  =  m  [27] 

When  these  functions  are  known,  the  motion  is  com- 
pletely known,  for  the  path  may  be  obtained  by  eliminating 


24  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

t  between  these  three  equations  taken  in  pairs,  which  will 
give  two  surfaces  whose  intersection  is  the  path,  or  equa- 
tions [27]  may  be  regarded  as  the  parametric  equations 
of  the  path.  The  position  of  the  point  on  the  path  is  also 
known,  for  as  soon  as  t  is  given  the  coordinates  of  the  point 
at  that  instant  are  determined  by  [27].  These  equations 
[27]  are  termed  the  equations  of  motion,  and  the  problem 
of  finding  the  motion  of  a  point  is  the  problem  of  finding 
these  equations. 

A  second  method  of  fixing  the  position  of  the  point  M 
in  its  path  is  the  following:  Its  distance  s  from  some  fixed 
point  Mo  on  the  path,  measured  along  the  path,  may  be  given 
at  each  instant,  the  direction  of  motion  being  chosen  as 
positive.    The  equation  of  motion  then  has  the  form 

^=m  [28] 

the  path  being  supposed  given, 

19.  Rectilinear  motion. — If  the  path  is  a  straight  line, 
the  motion  is  termed  rectilinear.  Choosing  the  path  as 
the  axis  of  X,  the  two  methods  of  fixing  the  position  of  the 
point  M  become  identical,  and  the  equation  of  motion  has 
the  form 

x-m  [29] 

the  point  Mq  being  the  origin  0. 

Now  for  a  given  value  of  t,  x  can  have  one  value  only 
and  hence  the  function  f{t)  is  uniform,  and  if  a  continuous 
sequence  of  values  of  t  be  substituted  in  [29]  we  must  get  a 
continuous  sequence  of  values  for  x,  since  M  cannot  pass 
from  one  point  on  its  path  to  another  without  passing 
through  the  intermediate  points.  This  same  equation  [29] 
permits  us  to  solve  the  inverse  problem,  that  is,  to  answer 


KINEMATICS  OF  A  POINT.     GENERAL  THEOREMS        25 

the  question*.  What  are  the  times  when  M  passes  a  given 
point  on  its  path?  In  this  case  x  is  given  and  we  require  t. 
It  is  necessary,  therefore,  to  solve  [29]  for  t,  which  we  may 
suppose  done,  with  the  result  that 

t  =  cf>(x)  [30] 

which  is  the  equation  of  motion  in.  its  second  form.  Often 
the  conditions  of  the  problem  lead  us  to  an  equation  of 
this  form,  which  must  be  solved  for  x  to  get  the  first  form. 
For  theoretical  considerations,  however,  the  first  form 
is  much,  to  be  preferred.  As  has.  already  been  remarked, 
j{t)  can  have  only  one  value-  for  a  given  value  of  t,  but  cf){x) 
may  have  more  than  one  value  for  a  given  value  of  x,  since 
the  moving  point  ma}'^  pass  the  same  place  in  its  path  any 
number  of  times. 

20.  Units  of  time  and  of  length. — As  has  already  been 
remarked  in  Art.  16,  time  is  expressed  in  mean  solar  seconds. 
The  unit  may  be  taken  as  the  second  itself  or  any  multiple 
thereof,  as  the  minute,  hour,  day,  or  year. 

The  unit  of  length  is  usually  the  metre  or  the  kilometre. 
I'nless  the  contrary  is  stated  the  second  and  metre  will 
be  used  as  the  units. 

21.  Case  where  it  is  necessary  to  specify  the  units. — In 
general  a  numerical  equation  of  motion 

x=/(0 

arbitrarily  given  has  no  meaning  unless  the  units  are  stated. 
Thus  the  equation 

x=<2 


26  ELEMENTS  OF'  KINEMATICS  AND  MECHANICS 

gives  ar  =  3600  metres  in  1  minute  if  the  second  and  metre 
are  adopted,  but  gives  x  =  l  kilometre  in  1  minute  if  the 
kilometre  and  minute  are  adopted. 

22.  Homogeneous  equations.  —  The  equations  which, 
instead  of  being  given  empirically,  express  laws  of  nature 
cannot  depend  on  the  choice  of  units  and  must  therefore 
be  homogeneous,   that  is,  are  necessarily  of  the  form 


f=/(|).  [311 


where  ).  is  a  length  and  t  a  time  expressed  in  the  same  units 

as  X  and  t,  and  hence  if  the  units  be  changed  the  equation 

X  t 

is  not  affected  since  the  ratios  —  and  —  remain  the  same. 

/  z 

23.  Change  of  units. — If  we  have  on  the  other  hand  a 
non-homogeneous  equation,  such  as  would  result  from 
numerical  observations, 

x=/(0 

with  determined  units  of  time  and  length,  it  is  easy  to  see 
what  this  becomes  when  other  units  of  time  and  length 

are    used.      Suppose   the    new  unit   of   length  to  be   ( j) 

of  the  old,  and  the  new  unit  of  time  I  - )  of  the  old.      Then 

it  is  clear  that  a  determined  length  which  contains  the  old 
unit  X  times  contains  the  new  one  Xx  times  and  its  new 
numerical  value  will  be 

Xx  =  kc,       whence        ^^y 


KINEMATICS  OF  A   POINT.     GENERAL  THEOREMS.       27 
Similarly  ti  =  Tt,         whence        t=— 


and  the  equation 
becomes 


=/(^ 


/I 


Consider  again,  as  an  example,  the  equation 


and  suppose  the  metre  and  second  to  be  the  units.     It  may 
then  be  written 


SO  that  at  the  end  of  a  minute 

re  =  3600™ 
Taking  the  kilometre  and  minute  as  new  imits, 


T  =  1000    and    -=60 

A  r 


28  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

«o  that  tHe  equation  becomes 

1000x1  =  (60^i)2  =  3600fi2 

or  xi  =  3.6fi2 

For  ii  =  l  we  have  a:i=3.6,  which  means  that  at  the 
end  of  1  minute  the  abscissa  of  the  moving  point  is  3.6""", 
which  ^  what  we  found  with  the  old  units.     The  equations 

t 


1km       I  iminy 


are  then  equivalent. 

24.  Uniform  rectilinear  motion.  Velocity.  —  Motion  is 
termed  uniform  rectilinear  when  the  path  is  a  straight  line 
and  equal  distances  are  passed  over  in  equal  times. 
Choosing  the  X-axis  as  the  path  and  counting  time  from 
the  instant  when  the  point  is  at  the  origin,  0,  its  position 
at  any  other  instant  is  given  by  its  abscissa.  Let  M  be 
its   position  at  the   instant  t  (fig.  13)   and  Mi  its  position 


Fig.  13. 


-^ 


at  the  instant  t  +  Jt,  At  being  positive.  The  geometric 
magnitude  MMi  has  the  algebraic  value  Ax,  where  this 
represents  the  increase  in  the  abscissa  x  of  the  moving  point 
in  time  At.    If  in  the  direction  MMi  a  vector  MV  of  length 


KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS      29 

—-   be  drawn,  this  vector  is,  by  definition,  the  velocity  of 
Jt 

the  uniform  motion  at  the  instant  t.  The  length  of  the  vec- 
tor MF  is  termed  the  algebraic  value  of  the  velocity,  or  the 
speed,  and  is  denoted  by  v.  It  is  positive  or  negative  accord- 
ing as  MV  is  directed  in  the  positive  or  negative  direction 
along  OX.     In  magnitude  and  sign,  then, 


Jx 


Since  the  motion  is  uniform,  that  is,  since  equal  distances 
are  passed  over  hi  equal  tunes, 

Jx  =  kJt 
no  matter  what  values  t  and  Jt  have,  and 

v=-n  =  k,  a  constant. 
Jt 

Hence,  the  velocity  in  uniform  rectilinear  motion  is  con- 
stant in  magnitude  and  direction. 

Remark. — If  Jt  =  l,  then  v  =  Jx,  whence  the  velocity  in 
uniform  rectilinear  motion  is  in  magnitude  and  direction 
the  space  travelled  in  unit  of  time. 

25.  Equation  of  uniform  motion. — Since  v  is  the  dis- 
tance travelled  in  unit  time,  the  distance  travelled  in  /  units 
is  vt,  and  if  xq  be  the  position  of  the  point  at  instant  t  =0,  its 
position  at  the  instant  t  is  given  by 

x=Xo  +  vt  [32] 


30  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

26.  Accelerated  rectilinear  motion. — Every  motion  not 
uniform  is  accelerated,  by  definition.  Let  the  accelerated 
motion  be  defined  by  x  =  f(t).  The  displacement  MMi  which 
the  point  undergoes  when  t  increases  by  Jt  is  a  vector  whose 
algebraic  value  is  Jx. 

Velocity. — If  in  the    direction  MMi   (fig.    14)   a  vector 

Ax 
MW  of  length  —    be  drawn,  this  vector,  whose  algebraic 


Jt 


-^ ^ 


0  M  M, 

Fig.  14. 

Ax 
value  is  --,  is  termed  the  mean  velocity  of  the  moving  point 

during  the  interval  Jt.  As  Jt  tends  to  zero,  this  vector  MW 
tends  to  a  limiting  vector  MV,  which  is  termed  the  velocity 
at  the  instant  t.  The  algebraic  value  of  MV,  termed  the 
sjpeed  at  instant  t,  is 

,.     Jx~\  dx 

v=lim— ■  =3- 

Jtjjt^o     ac 

From  the  equation  of  motion 

x=m 

we  have  by  differentiation 

.        "-t'fW  [33] 

27.  Numerical  examples. — A  mean  speed  is  thus  seen  to 
be  the  ratio  of  two  magnitudes  essentially  positive  and  of 


KINEMATICS  OF  A 'POINT.    GENERAL  THEOREMS.       31 

different  nature;  the  ratio  of  a  length  to  a  time.  These 
two  magnitudes  can  be  expressed  in  two  absolutely  inde- 
pendent units,  and  it  follows,  then,  that  the  numerical 
value  of  a  mean  velocity  does  not  have  any  meaning  unless 
the  units  are  specified,  for  the  same  velocity  can  be  expressed 
by  different  numbers  depending  on  the  units  chosen.  Sup- 
pose, for  example,  a  train  takes  five  hours  to  go  from  Wash- 
ington to  New  York.  The  distance  is  354  km.  (about), 
and  hence  the  mean  speed  between  AVashington  and  New 
York  is 

354 
Vm  =  £-  =70.8  km.  per  hour 

^^  „  km. 

=  70.8  r 

hour 

Expressing  the  distance  in  miles, 


-F"  =  44  miles  per  hour 


, ,  miles 
=44 


hour 


28.  Knots  and  nautical  miles. — The  speed  of  a  ship 
is  usually  expressed  in  knots.  This  is  an  abbreviation  of 
language,  and  the  units  here  understood  are  the  nautical 
mile  and  the  hour.  To  say  then  that  a  ship  has  a  speed  of 
20  knots  is  to  say  that  it  has  a  speed  of  20  nautical  miles 
per  hour,  or 

«^  naut.  miles 

V  =  20 r 

hour 


32  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

A  nautical  mile  is  the  length  of  1'  of  the  earth's  meridian 
or  about  1852  metres.  A  speed  of  one  knot  is  equivalent 
to  a  speed  of  .514  metres  per  second,  or 


naut.  mile      _     metres 
hour         ■       second 


29.  Equation  of  accelerated  motion  in  terms  of  the  mean 
velocity. — If  the  mean  velocity  during  any  interval  is  known, 
the  distance  travelled  is  at  once  obtained  from  the  relation 

Jx     Xi  — X 
which  gives 

Xi^X^-Vmih-t) 

so  that  knowing  the  position  of  the  point  at  the  instant  t 
and  its  mean  velocity  during  the  interval  t\—t  we  have  its 
position  at  the  instant  ^i. 

30.  Acceleration. — Uniform  rectilinear  motion  is  charac- 
terized by  the  property  that  at  each  instant  the  velocity 
is  the  same  in  magnitude  and  direction.  In  accelerated 
motion  the  velocity  varies  from  instant  to  instant,  and 
it  is  useful  to  take  account  in  an  exact  manner  of  this  varia- 
tion. The  notion  of  the  vector  acceleration  is  thus  obtained. 
Let  the  moving  point  be  at  M  (fig.  15}  at  the  instant  t,  and 

1 ji- ^ ^ ^ * > X 

0  M  Ml 

Fig.  15. 

have  there  a  velocity  MV.  At  instant  ^1  let  Mi  and  Mi\\ 
denote  its   position  and  velocity,  and  let  V\>v.      In  the 


KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS       33 

direction  MMi  construct  the  vector  MK  of  length 

Vi—v    Av 
h-t  ^Jt 

This  vector  MK  is  termed  the  mean  acceleration  during 
the  interval  h—L  Let  Jt  tend  to  zero,  then  MK  tends 
to  a  limiting  vector  MJ,  which  is  termed  the  acceleration 
at  the  instant  t. 

The  algebraic  value  or  magnitude  of  MJ  is 


/  =  lim 

V\ 

—  V' 

ti 

-t  _ 

h^t 

.    dv 

or 

J=& 

But  since 

dx 

we  have 

i- 

dH 
'dfi 

[34] 


[35] 


If  the  acceleration  J  is  the  same  for  all  points  of  the 
path,  the  motion  is  termed  uniformly  accelerated,  and  the 
change  in  the  velocity  is  proportional  to  the  time.  If  the 
velocity  decreases  with  the  time  the  acceleration  is  negative 
and  J  is  drawn  in  the  opposite  direction 

31.  Equation  of  uniformly  accelerated  motion. — Since 
in  this  case  J  is  constant  we  have 


3^  = ;-,  a  constant 


34  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

,  dx 

whence  -Tr  =  ft+c 


and  x-^^yt^  +  ct  +  Ci 

But  making  t  =  0  we  have 
^dx'' 


dtjo     *" 


and  Xo  =  c\ 

whence  x=Xo  +  Vot-\-^  yt^  [36  ] 

Where  Xq  and  Vq  are  the  displacement  and  velocity  at  the 
instant  t  =  0. 

32.  Discussion  of  the  motion  by  means  of  the   equation 
of  motion. — If  we  are  given  the  equation  of  motion 

the  position,  velocity,  and  acceleration  at  any  instant  may 
be  obtained  by  differentiating  this  equation  with  respect 
to  t,  for 


i=%=rit) 


Conversely,   if  we  are  given  the  acceleration  at  every 
instant,  that  is,  if  we  are  given  the  acceleration  as  a  func- 


KINEMATICS  OF  A  POINT.    GENERAL  THEOREMS        35 

Hon  of  t,  and  the  velocity  and  displacement  at  one  instant, 
we  can  determine  the  equation  of  motion.  In  order  to  do 
this  it  is  necessary  to  integrate  the  differential  equation 
of  the  second  order 

whose  solution  will  contain  two  constants  of  integration, 
such  that 

X  =  fit,C,Ci) 

But  c  and  Ci  may  be  determined  by  substituting  in 

X  =  f(t,C,Ci) 

dx 
and  v  =  -^  =  }'(t,c,ci) 

the  known  values  of  x  and  v  for  the  given  value  of  t. 

33.  Case  where  the  displacement  is  taken  as  independent 
variable. — It  is  often  useful,  instead  of  determining  the  cir- 
cumstances of  the  motion  at  each  instant,  to  determine 
them  at  each  point  of  the  path,  that  is,  to  express  them  in 
terms  of  the  displacement  instead  of  the  time.  This  is 
equivalent  to  taking  x  instead  of  t  as  the  independent  vari- 
able, and  expressing  the  equation  of  motion  in  the  form 

t  =  cl){x) 

This  may  be  given  directly  or  it  may  be  obtained  by 
solving  the  equation 

x=}{t) 
for  t. 


36  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

To  obtain  the  velocity  and  acceleration  in  terms  of  x, 
we  have 

^^      '         '  [37] 


dt     dt_     4>'{x) 
dx 

dH^ 

.    d^x_       dx^  rix) 

^    df         (diY^      4>'^^ 
\dx 


CHAPTER  III 
APPLICATIONS    TO    ORDINARY    RECTILINEAR    MOTIONS 

A.  Uniform  Motion 
34.  Properties  of  uniform  motion. — 35.  Problem. 

B.  Uniformly  Accelerated  Motion 

36.  Properties  of  uniformly  accelerated  motion.  —  37.  Remark. — 
38.  Equations  obtained  by  taking  the  space  described  as  inde- 
pendent variable. 

C.  Periodic  Motions 

39.  Equations  of  periodic  motion. — 40.  Equation  of  harmonic  motion. 
— 41.  Amplitude,  elongation,  frequency,  argument,  phase. — 
42.  Velocity  and  acceleration. — 43.  Discussion  of  the  motion. — 
44.  Influence  of  the  phase. 

A.  Uniform  Motion 

34.  Properties    of    uniform    motion. — ^The     equation     of 
uniform  motion  was  deduced  in  the  following  form: 

From  this  we  have  the  following 

Theorem  I. — In  a  uniform  motion  the  abscissa  of  the 
moving  point  is  a  linear  function  of  the  time. 

37 


463707 


38  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Conversely,  if  the  motion  is  uniform,  then  the  velocity 
is  constant,  or 

dx 
v  =  37  =  c,  a  constant 
at 


Now  any  equation  of  motion  of  the  form 

x^a  +  ht 


dx 
gives  v  =  ^  =  b 


from  which  follows  the 

Theorem  II. — Any  rectilinear  motion  whose  displace- 
ment is  a  linear  function  of  the  time  is  uniform. 

35.  Problem. — Given  the  instants  of  passage  of  the  uni- 
formly moving  point  at  two  points  on  its  path,  find  the  equa- 
tion of  motion. 

Let  Xo  and  xi  be  the  abscissas  of  the  points  Mo  and  Mi 
which  the  moving  point  passes  at  instants  ^0  and  ti. 

The  velocity  is  then 

_a;i— Xo 
ti  —to 

and  the  equation  of  motion,  being  linear,  is  of  the  form 


APPLICATIONS  TO  ORDINARY  RECTILINEAR  MOTIONS     39 

But  when   t=to,    x=xo,    and  hence 


and  the  equation  of  motion  becomes 

^=^»  +  (|^)(«-'o) 

B.     Uniformly  Accelerated  Motion 

36.  Properties   of   uniformly   accelerated  motion. — From 
the  equation  of  the  motion 

x  =  Xo+vot  +  ^irt^ 


dx 
we  have  v  =  -^  =  VQ  +  Yt 


We  observe,  then,  that  y,  being  constant,  cannot  change 
its  sign,  while  v  being  linear  in  t  can  change  its  sign  once 
only,  namely,  at  the  point  where 

vo  +  r^  =  0 
which  corresponds  to  the  instant 


«,=-2! 


40  ELEMENTS  OF  KINExMATICS  AND  MECHANICS 

and  hence  to  the  displacement 

which  gives  the  unique  point  in  the  path  beyond  which 
the  mobile  cannot  go.  The  mobile  can  traverse  only  that  por- 
tion of  its  path  which  lies  on  one  side  or  the  other  of  this 
point,  the  positive  side  if  ^  is  positive,  the  negative  side 
if  /-  is  negative,  since  for  t=  ±gc ,  x  =  +  oo  for  ^ > 0  and 
x=  —^  for  7'<0. 

The  general  properties  of  the  motion  are  then  contained 
in  the  following 

Theorem. — A  uniformly  accelerated  rectilinear  motion  has 
two  phages:  one  previous  to  the  unique  instant,  where  the 
velocity  vanishes  and  the  position  Mi  of  the  mobile  is  defined 
by  the  abscissa  (fig.  16). 


Fig.  16. 


2r 


the  other  subsequent  to  this  instant. 

During  the  first  phase  the  mobile  M  travels  toward  the 
point  Ml  with  a  uniformly  retarded  motion  and  reaches  it  at 
the  instant 


f  -     ^ 


There  it  comes  to  rest  and  then  returns  on  its  path  during  the 
second  phase,  describing  the  original  path  in  the  opposite  direc- 
tion with  a  uniformly  accelerated  motion. 


APPLICATIONS  TO  ORDINARY  RECTILINK^R  MOTIONS     41 

It  reaches  the  point  Mi  but  once,  every  other  point  in  the 
path  twice.  It  takes  the  same  time  to  go  from  any  point  M 
to  Ml  as  to  return  and  repasses  M  urith  th£  same  velocity  as 
at  first  passage  hut  in  opposite  direction. 

To  establish  this  theorem,  we  choose  the  positive  direc- 
tion of  the  X-axis  so  as  to  make  f  positive. 

Suppose,  then,  the  acceleration  positive.  Equation  [36} 
may  be  written 


whence 


dt  =  'V  +  j) 


Vo^  ^  Vo 

or,  supposmg         xi=Xo  —  7y~,        ^i= 


dx 

-j^  =  r(t-ti)=^-r(ti-t) 


These  equations  express  the  properties  indicated.  From 
t  —  ti  =  —  'X)  to  t  —  ti=0  the  mobile  comes  from  infinity  along 
the  positive  axis  of  X  and  approaches  the  point  Mi  of 
abscissa  xi,  which  it  reaches  at  the  instant  ti. 

When  t  —  ti  increases  from  0  to  +oo,  x—xi  takes  the 
same  values  as  before,  which  shows  that  the  mobile  retraces 
its  path. 

For  two  values  of  t  —  ti  equal  and  of  opposite  sign,  x—Xi 
takes  values  equal  and  of  same  sign,  which  shows  that  the 
mobile  takes  the  same  time  to  go  from  any  point  to  Mi 


42  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

(Ix 
as  to  return;    and  —  takes  values  equal  and  of  opposite 

sign,  which  shows  that  the  velocities  are  equal  and  opposite 
going  and  returning. 

37.  Remark. — We  have  studied  the  motion  from  t=  —00 

to   t=  +CC  . 

Suppose  we  wish  to  know  what  the  motion  is  during 
a  finite  interval,  from  ^o  to  t,  say. 

If   the   initial    instant   ^0   precedes   the   critical   instant 

ti=  — —,  that  is,  if  7'o  and  ;-  have  opposite  signs,  then  both 

phases  of  the  motion  are  comprised  in  the  interval  t  —  to, 
and  the  mobile  starting  from  its  initial  position  moves  -to  Mi 
and  returns  to  its  initial  position,  which  it  will  pass  unless 
stopped. 

38.  Equations    obtained   by    taking   the    space    described 
as  the  independent  variable. — The  equation 


dx 
squared  gives 


-=v-^vo+rt 


=Vo^  +  2r{x-xo) 

The  velocity  Vo  and  the  displacement  xo  refer  to  the  epoch 
t=0.    If  then  h  is  the  space  described  from  the  epoch  ^=0, 

i^^vo^  +  2rh  [39] 

— a  fundamental  equation. 


APPLICATIONS  TO  ORDINARY  RECTILINEAR  MOTIONS     43 


C.  Periodic  Motions 

39.  Equation  of  a  periodic  mbtion. — A  function  f(t) 
of  a  variable  t  is  said  to  be  periodic  when,  for  a  certain  positive 
increment  T  of  the  variable  t,  the  function  does  not  change 
its  value,  that  is,  when 

f(t  +  T)=f(t)  [40] 

whatever  be  t. 

If  T  is  the  smallest  increment  satisfying  this  condition, 
it  is  called  the  period  of  the  function  f(t). 

Since  [40]  is  true  whatever  be  t,  it  follows  that 

f{t±nT)=^f{t) 

where  n  is  any  positive  integer. 

Suppose  a  mobile  has  a  motion  represented  by  the  equa- 
tion 

f(t)  being  a  periodic  function  of  period  T.    It  follows  then 
that: 

If  the  mobile  at  the  instant  t  has  the  position  defined 
by  the  abscissa  x,  it  will  repass  this  same  position  at  the 
end  of  each  successive  interval  T  and  have  there  the  same 
velocity  and  acceleration. 

For  f(t  +  T)=^f{t) 

and  f(t  +  T)==f(t) 


44  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

which  shows  that  the  derivative  of  a  periodic  function  is 
also  periodic.  This  is  true  of  all  the  derivatives,  and  in 
particular  of  the  two  which  represent  the  velocity  and 
acceleration  of  the  mobile. 

40.  Equation  of  harmonic  motion. — A  periodic  motion 
which  is  particularly  important  in  mechanics,  and  in  all 
branches  of  physics,  is  that  in  which  the  displacement  varies 
as  the  sine  or  cosine  of  an  angle  which  is  itself  a  linear  func- 
tion of  the  time. 

The  equation  of  such  a  motion  is  then,  by  definition, 

a:  =  acos^l  (1) 

or  a:  =  asin^J  (2) 

where  ^  is  a  linear  function  of  the  time. 
Let 

d=nt-4> 

where  a,  n,  and  ^  are  constants. 

By  a  proper  choice  of  the  constants  both  (1)  and  (2) 
define  the  same  motion,  hence  it  is  necessary  to  consider 
one  of  them  only.     In  fact  if  we  put 


(1)  takes  the  form 

x  =  asin  di 

where  di  is  linear  with  respect  to  the  time. 


APPLICATIONS  TO  ORDINARY  RECTILINEAR  MOTIONS     45 

Again,  in  (2),  which  is  the  one  we  shall  retain,  we  can 
always  regard  n  as  positive,  for  if  n  is  negative,  then  putting 

di=-d=-nt  +  <J) 
(2)  becomes 

a;  =  a  sin  ^i 

in  which  the  coefficient  of  t  is  positive.     Also,  a  may  be 
regarded  as  positive,  for  if  it  is  negative,  then  putting 

ai=  —a 
we  have 

x  =  ai  sin  ^i 

where  oi  is  positive. 

Finally  ^  may  always  be  considered  to  lie  between  0 
and  27r,  since  if  it  is  not  contained  in  these  limits,  it  can 
be  brought  within  them  by  the  addition  of  a  suitable  posi- 
tive or  negative  integral  multiple  of  2?!,  which  will  not  affect 
the  value  of  x. 

We  can  then  write  the  equation  of  motion  in  the  form 

a:  =  a  sin  ^  =  o  sin(nt  —  ^)  [41] 

where  a  and  u  are  positive  and  0  lies  between  0  and  2;:. 

41.  Amplitude,  elongation,  period,  frequency,  argument^ 
phase. — Since  sin  6  varies  from  —1  to  +1  for  increasing 
values  of  t,  the  displacement  x  varies  between  —a  and  +a. 

If  then  on  the  path  X'OX  of  the  mobile  starting  from 
the  origin  0  we  lay  off  distances 

OAo=OAi=a 


46  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

on  each  side  of  the  origin  (fig.  17),  the  motion  consists  of 
an  infinite  sequence  of  oscillations  of  the  mobile  between 
Ao  and  Ai,  and  is  termed  a  simple  vibration  or  a  simple 
harmonic  motion.    The  length  a  is  called  the  amplitude, 


A,  0  Ao 

Fig.  17. 

and  the  displacement  OM  =  x  at  any  instant  is  called  the 
elongation  of  the  mobile  at  that  instant. 

The  period  T  of  the  motion  is  the  increment  which  must 
be  given  to  t  to  bring  x  back  to  its  same  value  in  magni- 
tude and  sign,  and  is  therefore  determined  by  the  relation 


sin  {nt  -cf>)  =  sin  [n(f  +  7")  -  <^] 

ich  gives 

nT  =  27: 

n 

The  number  of  oscillations  per  unit  of  time,  generally 

per  second,  is  called  the  frequency,  and  the  length  of  one 

.     1         n 
oscillation  being  T,  the  frequency  is  — ,  or  — . 

I  Lit 

The  variable  angle  d  is  called  the  argument,  and  the 
constant  angle  ^  the  phase. 
Putting  i=0  in  [41],  we  have 

a:o  =  a  sin  ( —  0) 

Whence  —  ^  is  the  value  of  0  at  the  initial  instant. 


APPLICATIONS  TO  ORDINARY  RECTILINEAR  MOTIONS     47 

42.  Velocity  and  acceleration. — The  velocity  and  accelera- 
tion are  given  by 


dx 
v  =  —r-  =  an  cos  6  =  an  cos  {nt  —  6)  [42] 

at 


j  =  -^= —an^sin6= —n^x  [43] 


Hence,  the  acceleration  is  proportional  to  the  elongation 
of  the  mobile  and  is  constantly  directed  toward  the  centre  of 
the  path. 

43.  Discussion  of  the  motion. — Suppose  the  phase  <j)  to 
be  zero,  which  is  equivalent  to  supposing  that  t  is  taken 
equal  to  zero  at  the  instant  when  the  mobile  passes  0.  The 
equation  of  motion  then  reduces  to 


.       ,  .    2;r« 

x  =  a  sin  nt  =  a  sin  — 


Whence  v  =  -:r  =  an  cos  nt^ns/a^—x^ 

dt 


d^x 
j  =  -n,=  —an^  smnt=  —n^x 
dt^ 


It  is  sufficient  to  discuss  an  oscillation,  that  is,  to  dis- 

T 

5S  the  motion  from  t=  —  —  to 

period  the  motion  is  reproduced. 


T  T 

cuss  the  motion  from  t=^——  to  t=-\--,  since  after  each 


48  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

T  T 

From  t=  ——  to   t=  +—     the    elongation    varies    from 
4  4 

— o  to  +a  and  the  mobile  describes  its  complete  path  in 
the  positive  direction  from  Ai  to  Aq.  At  Ai  when  x=  —a, 
the  velocity  is  zero,  and  increases  from  this  point  to  0,  where 
it  has  its  maximum  value  na.  From  A  to  ylo  it  decreases 
and  becomes  zero  again  at  ^o-  The  mobile  comes  instan- 
taneously to  rest  at  ^0  and  then  describes  its  path  AqAi 
in  the  negative  direction. 

We  observe  finally  that  during  a  complete  oscillation 
the  mobile  passes  twice  each  point  of  its  path,  once  in  the 
positive  direction  and  once  in  the  negative. 

If  6  is  the  argument  for  which  it  passes  a  given  point 
the  first  time,  the  argument  for  which  it  passes  it  the  second 
time  is 


If  then  it  passes  the  point  at  the  instant  t,  it  again  passes 
it  at  the  instant  h  determined  by 


27rfi  27:t 


T 
or  <i  =  2 — ^ 


The  expression  for  the  velocity 


dx  .  2nt 

-T-^an  cos  nt  =  an  cos-=- 
dX  1 


APPLICATIONS  TO  ORDINARY   RECTILINEAR  MOTIONS     49 

shows  that  at  the  instant  ti  it  has  the  same  velocity  as  at 
the  instant  t,  but  with  the  sign  changed. 

44.  Influence  of  the  phase. — We  have  supposed,  in  the 
discussion  of  the  motion,  the  phase  ^  to  be  zero.  But 
the  motion  retains  the  same  properties  whatever  the  phase. 
Imagine  two  mohiles  with  the  same  simple  harmonic  motion, 
but  different  phases,  that  is,  with  the  same  amplitude  and 
period,  but  one  with  phase  zero  and  the  other  with  phase  (f>. 

Representing  by  x  and  Xi  the  elongations  of  the  two 
mobiles  at  the  instant  t,  we  have 

x  =  a  sin  nt 

Xi=a  sin  (nt  —  <p) 

_      .  <f> 

Puttmg  to  =  ~,  the  last  equation  may  be  written 

X\=a^m.n{t—tQ) 

where  ^  is  the  phase  measured  in  angle  and  to  the  phase 
measured  in  time. 

If  then  for  the  second  mobile  we  measure  the  time  from 
the  instant 

n 

its  equation  of  motion  is  the  same  as  that  of  the  first  mobile, 

which  shows  that  its  motion  is  the  same  as  that  of  the  first, 

6 
but  it  passes  each  point  of  their  common  path  fo  =  ~  units 

of  time  after  the  first. 


50  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

The  two  mobiles  behave  as  two  trains  which  run  between 
the  same  two  cities,  the  second  to  units  of  time  after  the 
first.  They  have  the  same  motion  and  stop  at  the  same 
stations,  but  one  always  a  fixed  interval  of  time  ahead  of 
the  other. 


CHAPTER  IV 

RELATIVE  MOTION  ALONG  A  LINE 

45.  Relative  motion. — 46.  Equation  of  relative  motion. — 47.  Rela- 
tive velocity  and  acceleration. — 48.  Apparent  motion. — 49.  Con- 
vective  motion. — 50.  Simultaneous  motions. — 51.  Composition 
of  harmonic  motions  of  same  period. — 52.  Representation  of 
the  amplitude  and  phase  by  a  vector. — 53.  Equation  of  the 
resultant  motion. — 54.  Vibrations  "of  different  periods. 

45.  Relative  motion. — Consider  the  motion  of  two 
mobiles  along  the  same  line  X'OX  (fig.  18).    Let  Mc  and  Ma 

v' Mc  Ma 

X 5 ^ . X 

Fig.  18. 

represent  the  two  mobiles  and  .re  and  Xa  their  abscissas, 
or  displacements,  referred  to  the  origin  0.  Denote  the 
distance  McMa  by  ^rj  and  let  the  sign  of  $r  be  determined 
by  the  relation 

Cr  ^^  Xa      Xc  L^^J 

The  algebraic  value  of  <?r  then  determines  the  relative 
positions  of  Mc  and  Ma,  that  is,  it  determines  their  dis- 
tance apart  and  their  order  on  the  line  X'OX,  so  that  given 
the  position  of  one  of  them  the  position  of  the  other  can  be 

fil 


52  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

determined  mthout  ambiguity,  ^r  is  termed  the  relative 
displacement  of  Ma  vdth  respect  to  Mc. 

If  we  are  concerned  with  the  relative  positions  only  of 
two  mobiles,  as  of  two  runners  or  of  two  race-horses,  it  is 
necessary  to  know  f^  only.  In  distinction,  Xa  and  Xc  are 
called  the  absolute  displacements  of  Ma  and  Mc,  and  if  we 
require  the  absolute  positions  of  the  two  runners  on  the 
race-track,  then  Xa  and  Xc  must  both  be  known. 

Because  of  the  relation  [44]  which  connects  <fr,  Xa  and  Xc, 
it  is  sufficient  to  know  two  of  them  to  determine  the  third. 

The  motion  of  Ma  with  respect  to  Mc  is  termed  the 
relative  motion  of  Ma. 

46.  Equation  of  relative  motion. — If  Cr  be  known  as  a 
function  of  the  time,  then  at  any  instant  the  relative  position 
of  Ma  is  Imown.    This  equation 


-W) 


which  gives  the  relative  displacement  in  terms  of  the  time, 
is  known  as  the  equation  of  relative  motion  of  Ma  referred 
to   Mc. 

If  the  equations  of  absolute  motion  of  Ma  and  Mc  are 
known,  then  the  equation  of  relative  motion  is  at  once  deriv- 
able.   For  if 

Xa==fa{t) 

and  Xc=fc{t) 

then  e.==/a(0-/c(0  [45] 

47.  Relative  velocity  and  acceleration. — In  exactly  the 
same  manner  as  the  notion  of  absolute  velocity  was  obtained, 


RELATIVE  MOTION  ALONG  A  LINE  53 

we  can  get  the  notion  of  relative  velocity.  The  absolute 
velocity  of  Ma  may  be  regarded  as  the  rate  at  which  Ma 
is  receding  from  or  approaching  0,  and  the  relative  velocity 
of  Ma  the  rate  at  which  it  is  receding  from  or  approaching 
Mc.  The  relative  velocity  then  is  the  derivative  of  Cr  with 
respect  to  the  time.  Similarly  the  relative  acceleration  is  the 
derivative  of  the  relative  velocity  wdth  respect  to  the  time. 
Differentiating  [45],  we  then  have 

d$r    dxa    dxc 
dt      dt      dt 

or  Vr=Va-Vc  [46] 

Differentiating  a  second  time, 

dvr_d^$  __d^Xa    dHc 
'dl^~di^^~dfi~~di? 

or  jr=ja-h  [47] 

Hence,  the  relative  displacemeyit,  velocity,  and  acceleration 
of  a  mobile  Ma  referred  to  a  second  mobile  Mc  is  eqvM  to  the 
absolute  displacement,  velocity,  and  acceleration  of  Ma  minus 
the  absolute  displacement,  velocity,  and  acceleration  of  Mc. 

48.  Apparent  motion. — Imagine  an  observer  situated  on 
Mc  and  observing  the  motion  of  Ma.  The  quantities  which 
he  would  measure  would  be  exactly  the  relative  displace- 
ment, velocity,  and  acceleration  of  Ma,  and  he  would  there- 
fore know  only  the  relative  motion  of  Ma.  For  this  reason 
relative  motion  is  important  and  is  often  termed  apparent 
motion. 


54  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Suppose,  for  example,  one  train  to  folow  another.  A 
passenger  on  the  second  train  can  observe  whether  he  is 
gaining  or  losing  on  the  first,  and  if  he  has  means  for  measur- 
ing the  distance  between  the  two  trains,  can  tell  how  much 
he  is  gaining  or  losing  per  minute,  that  is,  he  can  determine 
the  relative  or  apparent  motion  of  the  first  train. 

49.  Convective  motion. — Transposing  the  terms  of  [46] 
and  [47],  he  have 

Va=Vc+Vr  [48] 

ja-ic+ir  [49] 

The  absolute  velocity  and  acceleration  of  Ma  are  then 
seen  to  be  made  up  of  two  terms,  one  due  to  the  motion  of 
Mc  itself  and  the  other  due  to  the  relative  motion  of  Ma 
referred  to  Mc.  If  we  imagine  Ma  to  be  invariably  con- 
nected with  Mc  and  carried  along  with  it  in  its  motion, 
the  absolute  velocity  and  acceleration  of  Ma  are  the  same  as 
those  of  Mc,  but  if  Ma  moves  with  respect  to  Mc,  an  additional 
term  is  introduced  due  to  this  relative  motion.  We  then 
divide  the  absolute  motion  of  M„  into  two  components, 
one  due  to  the  motion  of  Mc  itself  and  the  second  due  to 
the  relative  motion  of  Ma  referred  to  Mc.  This  first  com- 
ponent due  to  the  motion  of  Mc  is  called  the  convective 
motion  of  Ma. 

ivclocitij      I 
,      , .      \of  Ma  equals  the  sum  of 
acceleration ) 

\     velocity     ]        ,   ,        ,    •      {     velocity     ]     . 
the  convective  <        7      ^  •      r  and  the  relative  \        ,      !       ^  or 
( acceleration  )  ( acceleration  ) 

Ma. 

Suppose,  for  example,  a  man  walking  down  the  aisle 
of  a  moving  car.  His  absolute  velocity — referred  to  one 
of  the  stations  as  origin,  say — is  equal  to  the  sum  of  the 


RELATIVE  MOTION  ALONG  A  LINE  55 

absolute  velocity  of  the  car  and  his  relative  velocity  in 
the  car. 

A  man  owns  a  rifle  that  gives  a  velocity  of  a  mile  a 
second  to  the  bullet.  He  stands  on  the  rear  platform  of 
a  train  moving  a  mile  a  minute  and  fires  down  the  track. 
What  is  the  motion  of  the  bullet? 

Considering  the  direction  in  which  the  car  is  moving 
as  positive,  then  the  convective  velocity  of  the  bullet  is 


,    mile 


mmute 
and  its  relative  velocity  is 


„^  miles 


mmutes 
Hence  its  absolute  velocity  down  the  track  is 

.     ^^        ^^  miles 
"  minutes 

the  minus  sign  indicating  that  it  is  moving  in  a  direction 
opposite  to  that  of  the  train. 

50.  Simultaneous  motions. — Due  to  the  method  just 
described,  of  breaking  up  the  absolute  motion  into  the 
convective  and  relative  motion,  a  mobile  is  said  to  possess 
both  of  these  motions  simultaneously,  and  this  way  of  speak- 
ing is  justified  by  what  precedes.  It  is  to  be  understood, 
however,  that  it  is  only  an  abbreviation  of  language.  The 
method  given  for  obtaining  the  absolute  motion  from  these 


56  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

two    simultaneous    motions    is   termed   the   composition  of 
motions. 

Any  number  of  intermediate  mobiles  Mc„  Mc^ .  .  .  Mg^ 
may  be  imagined  and  the  relative  motion  of  each  may  be 
given  with  respect  to  the  one  that  precedes  it.  Thus  the 
motion  of  Mc^  may  be  given  with  respect  to  0  and  the 
motion  of  Mg^  with  respect  to  Mc, . . .  the  motion  of  Mc 
with  respect  to  Mc^_^,  and  finally  the  motion  of  M^  with 
respect  to  Mc  .    We  then  have 

^a  =  ^c+^12  +  ?23+    .  .  .    +f(„-l)v  +  ^r  [50] 

where   f(i-i)f  denotes  the  displacement  of  M^.  relative  to 
Mc._p  and  ^r  the  displacement  of  Ma  relative  to  Mc^. 
Differentiating 

Va  =  Vc  +  Vr^^  +  Vr,,+    .  .  .    +^V(,_i),  +  Vr  [51] 

Ja  =  jc  +  U,  +  jr,,   +    .  .  .    +yr(,_i),  +  Jr  [52] 

51.  Composition  of  harmonic  motions  of  same  period. — 
If  we  have 

Xc         =  Oc  sin  {nt  —  4>c) 
^12       =ai2sm{nt-cf)i2) 


then 


^(,_i),=a(,_i),  sin  (72f-</)(,_i)J 
^r         =  cir  sin  {nt  =  (pr) 

Xa=ac  sin  (nt  —  <pc)+  •  •  •  +ar  sin  (nt  —  (f>r)         [53] 


RELATIVE  MOTION  ALONG  A  LINE 


57 


52.  Representation  of  the  amplitude  and  phase  by  a 
vector. — In  order  to  study  this  resultant  motion  of  M^  it 
is  convenient  to  make  the  following  geometric  construc- 
tion. Draw  in  a  plane  a  line  OP  (fig.  19)  not  necessarily 
in  any  way  connected  with  the  line  along  which  the  har- 
monic motions  take  place. 


Fig.  19. 

From  0  as  origin  draw  vectors 

OAc 0.4(i_,), 

of  lengths 

o-c CKt-i)t 

making  angles 

with  the  line  OP,  measured  in  the  positive  direction. 


.OAr 


.Or 


68  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

53.  Equation   of    the   resultant    motion. — Construct   the 
geometric  sum  of  these  vectors 

6A==6A,+  .  .  .  +0^(,_i)i+  . . .  +0^ 

and  let  a  represent  its  length  and  <}>  the  angle  it  makes  with 
OP,  measured  in  the  positive  direction. 
Draw  an  axis  OZ  making  the  angle 

TT 

with  OP,  measured  in  the  negative  direction. 

Then  the  projection  of  OA  on  OZ  equals  the  algebraic 
sum  of  the  projections  of 

OA, 0^(,_i),- OAr 

on  OZ  for  all  values  of  t,  and  expressing  this  analytically, 
we  have 

a cos(  —  —nt+^j  =ao cos(-  —nt  +  cj)c)  +  . . . 

+  a(i_i)iCosf|-n«  +  <^(i_i)J+  .  .  .  +ar  cosf|-n<  +  ^,j    • 

which  may  be  written 

o  sin  (nt  —  (f))  =  ac  sin  (n<  —  ^c)  +  •  •  •  +  «(t-  i)i  sin  nt  —  <f)^i^  d^-) 
+ .  .  .  +ar  sin  (nt  —  (f)r) 

Comparing  this  equation  with  [53]  we  see  that 

Xa  =  a  sin  (nt  —  ^)  [54] 


RELATIVE  MOTION  ALONG  A  LINE  59 

Therefore,  the  resultant  motion  of  any  number  of  simul- 
taneous harmonic  motions  of  the  same  period  and  along  the 
same  line  is  a  unique  harmonic  motion  of  the  same  period 
as  the  component  motions  and  whose  amplitude  and  phase 
are  given  by  the  geometric  sum  of  the  vectors  representing  the 
amplitudes  and  phases  of  the  component  motions. 

Corollary. — The  relative  motion  of  two  mobiles  having 
simple  harmonic  motions  along  the  same  line  of  same  period 
is  a  simple  harmonic  motion  with  this  period,  whose  amplitude 
and  phase  are  represented  by  the  geometric  difference  of  the 
two  vectors  representing  the  amplitudes  and  phases  of  the 
component  motions. 

54.  Vibrations  of  different  periods. — The  problem  of  the 
composition  of  two  harmonic  motions  of  different  periods 
is  much  more  complex.  Suppose,  for  example,  that  we 
wish  to  find  the  resultant  motion  of  two  harmonic  vibrations 
of  periods 

2n  ,     27r 

—    and    — 
n  ni 

its  equation  is  of  the  form 

a:  =  a  sin  {nt  —  <f))+ai  sin  (ni^  — 0i)  [55] 

and  although  each  component  is  harmonic,  the  resultant 
motion  will  be  neither  harmonic  nor,  in  general,  periodic. 

It  will  be  periodic  but  not  harmonic  when  the  two  periods 
are  commensurable,  for  if  2r7t  be  the  common  measure  and 
we  put 

,2;r  ,  2;r 

n=k —         ni=A:i— 

T  r 


60  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

where  k  and  A:i  are  integers,  then  when  t  is  increased  byr 
the  arguments  of  the  two  terms  of  [55]  increase  by 

2k7z     and     2ki7: 

respectively,  so  that  t  is  a  period. 

If  we  wish  to  find  the  Hmits  of  elongation  by  equating  the 
velocity 

v=na  cos  (nf  —  9!))  +  niai  cos  {nit  —  (j)i) 

of  the  resultant  motion  to  zero,  we  have  the  transcendental 
equation 

na  cos  {nt  —  <}>)+  riiai  cos  {nit  —  <j>\)  =  0 
which  can  only  be  solved  by  approximation  or  graphically. 


CHAPTER  V 


VELOCITY  AND  ACCELERATION  IN  CURVILINEAR  MOTION 


A.  Velocity 

55.  Displacement. — 56.  Velocity. — 57.  Algebraic  value  of  the  ve- 
locity.— 58.  Projection  of  the  velocity  on  the  displacement. — 
59.  Geometric  representation  of  the  path. — 60.  Projection  of 
the  velocity  on  any  axis. 

B.  Acceleration 
61.  Acceleration. — Hodograph. — 62.  Projection  of  the  acceleration 
on  any  axis. — 63.  Projection  of  the  acceleration  on  the  tan- 
gent and  principal  normal. — 64.  Projection  of  the  acceleration 
on  the  binormal. — 65.  Condition  that  the  acceleration  be  con- 
stantly tangential  or  normal  to  the  path.  Condition  that  the 
acceleration  be  constantly  zero. — 66.  Composition  of  motions 
along  the  same  path. 

A.  Velocity 

55.  Displacement. — In  rectilinear  motion  the  abscissa 
of  the  mobile  was  spoken  of  as  its  displacement.  This 
notion  extended  to  motion  in 
any  path  leads  to  the  definition 
of  the  displacement  of  the  mobile 
M  with  respect  to  the  fixed 
point  0  (fig.  20)  as  the  vector 
OM. 

Denote  this  vector  bj'^  the 
letter  r.  The  displacement  is, 
then,  a  variable  vector  with  an 
arbitrarily  chosen  origin. 

61 


Fio.  20. 


62 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


56.  Velocity.  —  The    velocity    in    curvilinear    motion    is 
defined  as  the  geometric  derivative  of  the  displacement,  or 


Dt 


[56] 


The   mobile  being  at  M  at  the  instant  t,  and  at  M\  at 
the  instant  t  +  M  (fig.  21),   then 


Fig.  21. 


y  =  lim 


r\  —r 


J<  =  0 


M   J 
=  lim  M]V]^t=^Q 
=  MV 

where  MW  is  a  vector  in  the  direction  MMi  of  length 

MMi 

At 

and  MV  is  the  limiting  position  of  MW  as  At  tends  to  zero, 
and  is  therefore  tangent  to  the  path  C  at  M. 


CURVILINEAR  MOTION :  VELOCITY  AND  ACCELERATION    63 

Hence,  the  direction  of  the  velocity  is  along  the  tangent 
to  the  path  at  M,  in  the  direction  of  motion. 
Now  [56]  may  be  written  algebraically 

TT     ,.     chord  MMi  1 


'       ""^          M 

Ji<-0 

P                    chord  MAfi  _  chord  MAfi 
Jt                arc  MMi 

arc  MMi 

M 

and  therefore 

/chord  MMi    arc  MMi\ 

^  =  ^''^[    arc  M Ml  Jt      )  j,=o 


..     chord  ikf  Ml    ,.      arcMMi 
=hm  ,^,,     .  hm 


=lim 


since  hm 


arc  MMi 

Tt 

chord  MM 


arc  MM 


-]   =1- 

1    -iMj-Af 


The  algebraic  value  of  the  velocity  may  then  be  regarded 
as  the  limit  of  the  ratio  of  the  distance  travelled  by  the  mobile 
to  the  time  required  to  travel  this  distance,  as  the  time  tends 
to  zero. 

This  is  analogous  to  the  velocity  in  rectilinear  motion. 

57.  Algebraic  value  of  the  velocity. — If  the  equation  of 
motion  along  the  given  path  c  be 

S='<f>(t) 


64  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

where  s  is  the  length  of  the  arc  MqM  (fig.  22)  measured 

M 

-V 


Fig.  22. 


from  some  fixed  point  Mq  on  the  path,  then  the  algebraic 
value  of  the  velocity  is 


74;-,'(0 


[57] 


for 


7=lim 


arc  MqM 

M 


ds 


58.  Projection  of  the  velocity  on  the  displacement. — The 
projection  of  F  on  r  is  the  projection  of  the  geometric  de- 
rivative of  a  vector  on  the  vector  itself,  and  hence  by  [26] 


K  = 


dr 
Jt 


If  then  the  equation  of  motion  along  the  given  path  be 
written 

r=/(0 


^-'t=/'(^) 


[58] 


CURVILINEAR  MOTION :  VELOCITY  AND  ACCELERATION     65 

59.  Geometric  representation  of  the  path. — The  displace- 
ment r  will  describe  a  cone  with  vertex  at  0,  and  the  path 
c  will  be  a  curve  lying  on  this  cone. 

It  may  be  imagined  as  traced  by  a  mobile  which  moves 

along  r  with  the  velocity  -17   while  r  describes  the  cone. 

dv 
If  -J—  is  zero,  the  path  is  the  intersection  of  the  cone  with 

a  sphere  of  radius  r. 

60.  Projection  of  the  velocity  on  any  axis. — From  the 
theorem  of  Art.  12  it  follows  that 

The  projection  of  the  velocity  on  any  axis  through  0,  as 
OX  (fig.  23),  is  the  algebraic  derivative  of  the  projection  of 
the  displacement  r  on  OX. 


Hence 


FiQ.  23. 


Fx  =  -j:(rcos^) 


or  V.=^-rsm»^  [59] 


66 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


B.  Acceleration 

6i.  Acceleration. — If  the  velocity  is  not  constant  in 
magnitude  and  direction,  it  is  necessary,  in  order  to  deter- 
mine the  motion,  to  know  how  it  varies. 

The  geometric  derivative  of  the  velocity  is  defined  to 
be  the  acceleration.  The  acceleration  is  then  a  vector  MJ 
(fig.  24),  the  geometric  derivative  of  the  vector  MV. 


Fig.  24. 


We  may  then  write 


/- 


DV 
Dt 


[60] 


Hodograph. — If  from  a  fixed  point,  0  say,  a  vector  be 
drawn  equipollent  to  MV,  the  path  described  by  the 
extremity  of  this  vector  is  termed  the  hodograph  of  the 
motion. 

Hence,  the  velocity  of  the  point  which  describes  the  hodo- 
graph is  equipollent  to  the  acceleration  of  the  mobile. 


CURVILINEAR  MOTION :  VELOCITY  AND  ACCELERATION     67 

62.  Projection  of  the  acceleration  on  any  axis. — Since 
(Art.  12)  the  projection  of  the  geometric  derivative  of  a 
moving  vector  on  any  line  is  the  algebraic  derivative  of 
the  projection  of    the  vector  on  that  line,  it  follows  that 

The  projection  of  the  acceleration  on  any  line  is  the  alge- 
braic derivative  of  the  projection  of  the  velocity  on  that  line, 
and  is  therefore  the  second  algebraic  derivative  of  the  projec- 
tion of  the  displacement  on  that  line, 

63.  Projection  of  the  acceleration  on  the  tangent  and 
principal  normal. — Since  the  projection  of  the  velocity  on 
the  tangent  at  M  is 

ds 
dt 


then  the  projection  of  the  acceleration ,  on  this  tangent  is 
(fig.  25) 


[61] 


Fig.  25. 


Since  the  principal  normal,  MN,  is  perpendicular  to  the  vec- 
tor OV;  then  (Art.  12,  Corollary)  the  projection  of  /  on  the 


68  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

principal  normal,  taking  the  inward  direction  as  positive,  is 


'--^itU 


where  ^  is  the  angle  between  MN  and  OV. 
This  may  be  written 

'^^        ^\ds  dtj^^^. 

-11 

~  P 

where  p  is  the  radius  of  curvature  at  M,  since 


[62] 


m 


1         1  . 

,    ,         =-    and  is  negative. 

ds  I  ^=90°      p 


64.  Projection  of  the  acceleration  on  the  binomial. — Let 

^  be  the  angle  between  the  binormal  MB  and  OY.    Then 
the  projection  J  on  MB  is 

\ai  /^=9o° 
^dcl)  ds 


ds   dt  /  0=90* 
=  0 


since  (  t~  =0 


\dsj^ 


Hence,  (he  acceleration  lies  in  the  osculating  'plane  and  is 
directed  toward  the  concave  side  of  the  path. 
It  is,  therefore,  as  drawn  in  fig.  25. 


CURVILINEAR  MOTION :  VELOCITY  AND  ACCELER AT  10 N     69 

65.  Condition  that  the  acceleration  be  constantly  tangen- 
tial or  normal  to  the  path. — ^Theorem  I. — In  order  that  the 
acceleration  he  constantly  tangent  to  the  'path  it  is  necessary 
and  sufficient  that  the  motion  he  rectilinear.  For  by  the 
first  problem  of  Art.  14  the  orientation  of  the  velocity  must 
be  constant. 

Theorem  II. — The  necessary  and  sufficient  condition  that 
the  acceleration  he  constantly  normal  to  the  path  is  that  the 
motion  he  uniform. 

For  by  problem  two  of  Art.  14  the  velocity  must  be 
constant  in  magnitude. 

Condition  that  the  acceleration  be  constantly  zero. — 
Theorem  III. — hi  order  that  the  acceleration  he  constantly 
zero  it  is  necessary  and  sufficient  that  the  motion  he  uniform 
rectilinear. 

For  by  problem  three  of  Art.  14  the  velocity  must  be 
constant  both  in  orientation  and  magnitude. 

66.  Composition  of  motions  along  the  same  path. — 
All  that  has  been  said  in  Chapter  III  on  uniform,  uniformly 
accelerated,  and  periodic  motions  in  a  straight  line  applies 
here  by  changing  x  into  s  and  acceleration  into  tangential 
acceleration. 

Similarly,  all  that  has  been  said  concerning  relative 
motion  along  the  same  straight  line  in  Chapter  IV  applies 
here  to  relative  motion  along  the  same  path  by  making 
the  same  changes. 


CHAPTER  VI 

ANGULAR  AND   AREAL  MOTION.     EQUATIONS   AND 
GENERAL  THEOREMS 

67.  Angular  displacement. — 68.  Angular  velocity. — 69.  Relation 
between  linear  and  angular  velocities. — 70.  Angular  accelera- 
tion.— 71.  Relation  between  linear  and  angular  accelerations. — 
72.  .\real  velocity. — 73.  Extension  of  theorems  already  found. — 
74.  Equations  of  motion.  Remark.  Units  of  time  and  dis- 
placement.    Homogeneity. — 75.  Equation  of  uniform  motion. — ■ 

76.  Equation     of     uniformly     accelerated     angular    motion. — 

77.  Equation  of  motion  when  angular  acceleration  is  not  con- 
stant.— 78.  Properties  of  uniformly  accelerated  angular  motion. — 
79.  Periodic  angular  motion.  Harmonic  motion. — 80.  Relative 
angular  motion. — 8L  Composition  of  harmonic  angular  motions 
of  same  period. 

67.  Angular  displacement. — Consider  the  case  where  the 
path  of  the  mobile  is  a  plane  curve,  and  let  0  (fig.  26)  be  any 
point  in  the  plane  of  the  path  and  OX  a  fixed  axis  in  that 
plane. 

I^et  the  mobile  be  at  M  at  the  instant  t,  and  denote  the 
angle  XOM  by  6. 

The  angular  displacement  of  the  mobile  about  0  at  the 
instant  i  is  a  vector  0  6  perpendicular  to  the  plane,  of  length 
6,  and  so  drawn  that  an  observer  standing  with  his  feet  at 
0  and  head  at  S  sees  the  mobile  moving  from  left  to  right 
in  its  path. 

70 


ANGULAR  AND  AREAL  MOTION 


71 


68.  Angular  velocity. — ^The  angular  velocity  of  the  mobile 
about  0  is,  by  definition,  the  geometric  derivative  of  the 


Fig.  26. 


angular   displacement,   and   denoting   the  angular   velocity 
by  (o,  we  have 


-    DO 


[63] 


Since  the  angular  displacement  remains  perpendicular 
to  the  plane  of  the  path,  the  angular  velocity  is  also  perpen- 
dicular to  this  plane,  and  is  equipollent  to  the  vector  Oco 
drawn  at  0  normal  to  the  plane  and  of  length  equal  to  the 

,  DS 
arithmetic  value  of  -jrr. 

Now  since  the  geometric  derivative  of  the  angular  dis- 
placement is  constantly  directed  along  the  displacement, 
its  magnitude  is  equal  to  its  projection  on  the  displacement, 

which  is  :j7. 
at 


72  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Therefore,  the  algebraic  value  of  the  angular  velocity  is 

dd 


io=—  [64] 


dt 


69.   Relation  between   linear  and    angular    velocities. — 

Making   use   of   the   theorem    of   Art.  12,  we  have  at  once 

V  cos,  (j)  = -T.  cos  (j>  =-T.{r  cos  d)  [65] 

where  ^  is  the  angle  between  the  linear  velocity  and  OX. 
Corollary. — If  the  path  is  a  circle  with  centre  at  0,  then 

<i>  =  0-\--^,  and  r  is  constant,  whence 

ds  .     dd 

—  -,-  sin  ^  =  r  sin  o-rr 

dt  dt 

T.     ds      dd 

y  =  ^  =  r^^r.  [66] 


Hence,  the  algebraic  value  of  the  linear  velocity  equals  the 
radius  of  the  circle  times  the  algebraic  value  of  the  angular 
velocity. 

70.  Angular  accelerations. — The  angular  acceleration  is, 
by  definition,  the  geometric  derivative  of  the  angular 
velocity.     Denoting  it  by  a,  we  have 

Since  the  angular  velocity  is  constantly  normal  to  the  plane 
of  the  path,  the  angular  acceleration  is  also,  and  its  alge- 


ANGULAR  AND  AREAL  MOTION  7a 

braic  value  equals  its  projection  on  the  angular  velocity. 
Hence 


dt  ~dt^ 


a  =  ^  =  ^  [681 


dW 
The  vector  Oa,  of  length  -77^,  normal  to  the  plane  is  then 

equipollent  to  the  angular  acceleration. 

71.  Relation  between  linear  and  angular  accelerations.— 
Denoting  by  4>  the  angle  between  the  linear  acceleration 
and  velocity,  and  using  the  theorem  of  Art.  12,  we  have 
from  [65] 

dV    d  {     1     d  ^  ) 

'^''^'^-dt-dti^^dfi'''''^  t«9J 

Corollary. — If  the  path  is  a  circle  with  centre  at  0, 
then 

..      dd 
^^'dt 

dt  "    dt 
and  J  cos  (I>=rjp2 

or  Jt  =  r^  =  ra  [70  J 


Hence,  Gie  tangential  component  of  the  acceleration  of  a 
mobile  moving  in  a  circle  is  equal  to  the  product  of  the  radius 
and  the  angular  acceleration  of  the  mobile  about  the  centre. 


74  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

The  normal  component  is 

T  =  —  =  — 

"  n  —  — 

p       r 

and  is  directed  toward  the  centre, 

72.  Areal  velocity. — Let  Mo  be  the  point  where  OX  inter- 
sects the  path   (fig.   27).     The  areal  displacement,   velocity, 


Fig.  27. 


and  acceleration  are  defined  in  exactly  the  same  wav  as 
the  angular  displacement,  velocity,  and  acceleration,  except 
that,  instead  of  the  angle  6,  the  area  A  of  the  curvilinear 
triangle  MqOM  is  used. 


Hence 


and 


Areal  T" 


dA 

''  dt 


Areal  /  = 


d  Areal  V     d^A 


dt 


dV^ 


Since,  when  6  and  the  path  are  given,  A  is  determined, 
these  can  be  expressed  in  terms  of  the  angular  velocity  and 
acceleration. 


ANGULAR  AND  AREAL  MOTION  76 

Let  dA  represent  the  increment  of  A  in  time  dt, 

Ad  ^        Ad 

then  either  7^^?r-  -AA'S.  -ri^—- 

2r.  -       -        2tz 


Ad  Ad 

or  T^^TT  ^AAt.  rri^— - 


r2  Ad  ^AA.Tx^  Ad 
Whence  YTi^^i^^'Tt 


or 


Now  when  At  tends  to  zero  the  two  variables  between 

A  A  r^  dd 

which  —rr  hes  tend  to  the  same  limit  ^r  ir 

At  2  at 


hence 


dAr^  dd 
dt''2   dt 


r2  dd 
or  ArealF  =  2-^  [71] 


d  Ir^  dd\ 
and  Areal./-^(2-dr)  t^^! 


These  formulae  find  immediate  application  in  the  motion 
of  a  planet  about  the  sun,  the  law  of  motion  being  such 
that  the  areal  acceleration  is  zero,  and  hence  the  areal 
velocity  constant. 


76  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

To  express  Areal  V  and  Areal  J  in  terms  of  the  rec- 
tangular coordinates  of  Mi  we  have 

X  =  r  cos  6        y  =  r  sm  6 

dx  dr  ^  .  jid 
-TT  =  TT  cos  0  —  r  sm  u^- 
dt     dl  dt 

dy    dr   .    ^  dd 

"77  =  ^T  sin  c/  +  r  cos  d~rr 
dt     dt  dt 

dy      dx      Ad 
Hence  "^dt-yW^'dl 


Therefore 


Areal  F4(.|-,f)  [73] 

1  /  d'^7j      d'^x\ 
and  Areal  J  =  ^  ^x-^  -  y^)  [74] 

73.  Extension  of  theorems  already  found.  —  What  has 
been  said  in  Chapters  III  and  IV  about  motion  along  a 
straight  line  can  for  the  most  part  be  extended  to  angular 
motion  about  a  point  by  reading  "  angular  displacement," 
"  velocity/'  and  "  acceleration  "  where  linear  displacement, 
velocity,  and  acceleration  occur. 

74.  Equation  of  motion, — As  in  the  case  of  rectilinear 
and  curvilinear  motion,  the  equation  of  the  path  being 
supposed  given,  in  the  form,  saj^, 

r  =  F{d) 
the  equation  of  motion  may  be  written 


ANGULAR  AND  AREAL  MOTION  77 

and  if  this  be  known  the  angular  velocity  and  acceleration 
are  immediately  obtained  from  the  relations 

If  the  angular  displacement  be  taken  as  the  independent 
variable,  the  equation  of  motion  may  be  written  in  the 
second  form 

t  =  <f>(d) 

and  the  velocity  and  acceleration  are  by  the  same  reasoning 
employed  in  Art.  33 

1 


(0  = 


a=  — 


Remark:  Units  of  time  and  displacement.  Homoge- 
neity.— ^The  unit  of  time  is,  unless  otherwise  stated,  the 
second,  and  the  unit  of  angular  displacement  may  be  taken 
as  the  degree  or  radian,  preferably  and  usually  the  latter 
unless  the  contrary  is  stated. 

The  same  remarks  as  made  in  Arts.  22  and  23  hold  con- 
cerning homogeneity  and  change  of  units. 

75.  Equation  of  uniform  motion. — ^As  before,  the  motion 
is  said  to  be  uniform  when  the  angular  velocity  is  constant, 
and  the  equation  of  such  motion  is 

d  =  do  +  (d  [75] 


78  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

obtained  by  integrating 

dd 

TT  =  w,  a  constant 

at 

The  two  theorems  of  Art.  34  now  read: 

Theorem  I. — In  a  uniform  angular  motion  the  angular 
displacement  is  a  linear  function  of  the  time. 

Theorem  II. — Every  angular  motion  whose  angular  dis- 
placement is  a  linear  function  of  the  time  is  uniform. 

76.  Equation  of  uniformly  accelerated  angular  motion. — 
By  integrating 

dW 
-^:^  =  (x,  a  constant 

we  arrive  at  the  equation 

d=do  +  (oot  +  ^at^  [76] 

which  is  the  equation  of  motion  when  the  angular  accelera- 
tion is  constant. 

77.  Equation  of  motion  when  the  angular  acceleration 
is  not  constant. — ^The  discussion  of  Art.  32  leads  at  once  to 
the  form 

e=f{t,c,ci)       . 

and  the  same  remarks  apply. 

78.  Properties  of  uniformly  accelerated  angular  motion. — 

The  entire  discussion  of  Arts.  36-38  inclusive  will  apply 
with  the  changes  noted  above,  and  give  the  fundamental 
equation 

(J^  =  (oo2+2aid-eo)  [77] 

and  the 


ANGULAR  AND  AREAL  MOTION  79 

Theorem. — A  uniformly  accelerated  angular  motion  has 
two  phases:  one  previous  to  the  unique  instant  where  the  velocity 
vanishes  and  the  position  Mi  of  the  mobile  is  defined  by  the 
displacement 


the  other  subsequent  to  this  instant. 

During  the  first  phase  the  mobile  travels  toward  the  point  Mi 
with  a  uniformly  retarded  angular  motion,  and  reaches  it  at 
the  instant 


tl— 

a 


There  it  comes  to  rest  and  then  returns  on  its  path  during  the 
second  phase,  describing  the  original  path  in  the  opposite 
direction  with  uniformly  accelerated  angular  motion. 

It  reaches  the  point  Mi  but  once,  every  other  point  in  the 
path  twice.  It  takes  the  same  time  to  go  from  any  point  M  to 
Ml  as  to  return,  and  repasses  M  with  the  same  angular  velocity 
as  at  first  passage  but  in  the  opposite  direction. 

79.  Periodic  angular  motion.  Harmonic  motion. — Har- 
monic angular  motion  being  defined  in  the  same  way  as 
harmonic  linear  motion,  its  equation  is 

d =13  sin  (nt-(f>)  [78] 

and  amplitude,  elongation,  period,  frequency,  argument,  and 
phase  have  the  same  meaning  as  in  Art.  41,  where  angle  is 
read  for  distance. 


80 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


The  entire   argument  of  Arts.  42-44   inclusive   applies^ 
giving  the  period 


and  frequency 


T  = 


n 


1     n 


angular  velocity  and  acceleration 


dd 
(i>  =  -Tr  =  ^n  cos  {iit—j)) 


a  = 


-n^d 


[79] 
[80] 


The  mobile  vibrates  about  the  centre  of  rotation  0 
(fig.  28),  coming  to  rest  at  the  points  Aq  and  Ai  and  return- 
ing on  the  path,  repassing  any 
point  with  the  same  angular 
velocity  but  in  opposite  direc- 
tion. The  phase  has  the  same 
influence  as  in  rectilinear  har- 
monic motion. 

8o.  Relative  angular  mo- 
tion.— All  the  statements  and 
theorems  of  Chapter  IV  on 
relative  motion  along  a  line 
apply  to  angular  motion  if  the 
terms  be  properly  interpreted. 
Suppose  two  mobiles  Mc  and  M„ 
to    be    moving    about   the    same    centre   0    (fig.    29),   the 


Fig.  28. 


ANGULAR  AND  AREAL  MOTION 


81 


angle  dc,  which  is  the  absolute  angular  displacement  of  Mc, 
is  termed  the  convective  angular  displacement  of  Ma,  and  ^r 
is  termed  the  relative  angular  dispkLcement  of  Ma. 


Fig.  29. 


We  have  then 


^r=Oa-dc 


d-&r      dda       ddc 

ir^ii  It 

d^^r      dWa       dW, 


""^^dr^ir—dT^ '"''-''' 


"'     dt^       dt^      dt^     ""    "' 


[81] 
[82] 

[83] 


The  equation  of  relative  angular  motion  is 

^r  =  fa{t)-fc(t)  [84] 

The  equations  of  Art.  50  for  the  composition  of  motions 
become 

6a  =  0,  +'Sl2  +   .  .  .    +\-l).  +  ^r  [85] 


0Ja  =  aJc+(Or,,+   .  .  .   +^r(,_i). 


+  COr 


«a  =  «r  +  «r„  +«r,3+    •  •  •    +^r(^,-l),  +  ^r 


[86] 

[87] 


82  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

8i.  Composition   of  harmonic   angular  motions  of  same 
period. — The  equations  of  Arts.  51,  52,  and  53  become 

/9c  =  /?c  sin  {nt-(j)c) 
%,=-t^i2  sin  {nt -(pi2) 


iL^r  =  ^r  sin   (Wi  —  (f)r) 

da  =  ^c  sin  {nt-4>c)+  ■  ■  •  +/?r  sin  {nt-^^ 
=  /?  sin  int  —  (j)) 

and  the  same  discussion  as  in  Art.  54  applies  to  composition 
of  angular  harmonic  motions  of  different  periods. 


CHAPTER  YII 

MOTION  REFERRED  TO  COORDINATE  AXES 

82.  Important  remark. — 83.  Projection  of  the  motion  on  a  plane 
and  on  an  axis. — 84.  Projection  of  the  velocity  and  accelera- 
tion.— 85.  Equations  of  motion  referred  to  the  coordinate 
axes. — 86.  Equations  of  the  projected  motion  on  the  coor- 
dinate planes. — 87.  Projection  of  the  path. — 88.  Equations  of 
the  projected  motion  on  the  coordinate  axes. — 89.  Projections 
of  the  velocity. — 90.  Equations  of  motion  of  the  point  which 
describes  the  hodograph. — Gl.  Projections  of  the  acceleration 
on  the  coordinate  axes. — 92.  Resume. — 93.  Most  general  mo- 
tion in  which  the  projected  motions  are  uniformly  accelerated. — 

94.  Motion   in    which   the    projected  motions  are  harmonic. — 

95.  Rectilinear  harmonic  motion  considered  as  the  projection 
of  uniform  circular  motion. 

82.  Important  remark. — ^Throughout  this  chapter  the 
axes  will  be  assumed  rectangular  and  the  projections  orthog- 
onal, however,  most  of  the  statements  and  theorems  will  be 
true  of  oblique  axes  with  the  necessary  modifications. 

83.  Projection  of  the  motion  on  a  plane  and  on  an  axis. — 
Consider  (fig.  30)  a  mobile  describing  in  space  any  path 
whatever  and  a  fixed  plane  77.  Let  M  be  the  position  of 
the  mobile  at  the  instant  t.  Project  M  on  77  into  m.  To 
each  position  of  M  in  its  path  will  correspond  a  determinate 
point  m.  The  point  m  may  then  be  regarded  as  a  second 
mobile,  whose  motion  is  completely  defined  by  that  of  M. 

83 


84 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


The  motion  of  m  is  termed  the  projection  of  the  motion  of 
M  on  the  plane  TI. 

Let  X'X  be  a  fixed  axis — not  necessarily  in  any  way 
connected  with  the  plane  77 — ^and  project  M  on  this  axis 
into  m' .     Regarding  this  point  m'  as  a  mobile,  its  rectilinear 


Fig.  30. 

motion  is  termed  the  projection  of  the  motion  of  M  on  the 
axis  X'X. 

84.  Projection  of  the  velocity  and  acceleration. — The 
velocity  of  M  is  the  geometric  derivative  of  the  displace- 
ment r, 


or 


^     Dt 


Hence  by  the  theorem  of  Art.  12,  that  the  projection  of 
the  geometric  derivative  of  a  vector  on  a  plane  is  equipol- 
lent to  the  geometric  derivative  of  the  projected  vector, 
we  have 


■    \Dtl. 


Dr^ 

Dt 


=v 


[88] 


from  which  follows  the 


MOTION  REFERRED  TO  COORDINATE  AXES  85 

Theorem  I. — The  projection  of  the  velocity  on  any  plane 
is  equipollent  to  the  velocity  of  the  projected  motion  in  tMt 
plane. 


Again,  from 


-   DV     Dh 
'^~  Dt  ~Dt^ 


we  have,  by  applying  the  theorem  of  Art.   12, 
—     (W\      Dv     . 

whence  the 

Theorem  II. — The  projection  of  the  acceleration  on  any 
plane  is  equipollent  to  the  acceleration  of  the  projected  motion 
in  that  plane. 

If  instead  of  projections  on  the  plane  77  we  consider  pro- 
jections on  the  axis  OX  the  same  reasoning  gives. 

Theorem  III. — The  projection  of  the  velocity  on  any  axis 
is  equal  to  the  velocity  of  the  projected  motion  on  that  axis. 

Theorem  IV. — The  projection  of  the  acceleration  on  any 
axis  is  equal  to  the  acceleration  of  the  projected  motion  on 
that  axis. 

85.  Equations  of  motion  referred  to  the  coordinate  axes. — 
Let  X,  y,  z  be  the  coordinates  of  the  mobile  at  any  instant 
t.  Then  x,  y,  and  z  are  uniform,  and  continuous  functions  of 
the  time. 


Let  x=f  (t) 

y=fx{t) 

2=/2(0 


[90] 
These   three   equations   taken   together   are   called   the 


86  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

equations  of  motion,  since  at  each  instant  they  define  the 
position  of  the  mobile  uniquely. 

But  any  two  of  them  toget^her  and  any  one  of  them 
have  interpretations  which  it  is  useful  to  know. 

86.  Equations  of  the  projected  motion  on  the  coordinate 
planes. — Suppose  the  motion  projected  on  the  XY  plane. 
The  projected  mobile  has  in  this  plane  at  each  instant 
the  same  coordinates  x  and  y  as  the  mobile  in  space.  Its 
equations  of  motion  are  then 

x=m        y=h(t)        2=0 

that  is,  the  two  equations 

x  =  m        y=h{t) 

are  the  equations  of  the  projected  motion  on  the  Xy-plane. 
Similarly  the  pairs  of  equations 

y=hit)      z=j2{t) 

and  3:  =  /(i)  2=/2(0 

are  the  equations  of  the  projected  motion  in  the  other  two 
coordinate  planes. 

87.  Projection  of  the  path. — Since  the  two  equations 

■      x=j{t)        i/=/i(0 

considered  in  the  XF-plane  are  the  equations  of  the  pro- 
jected motion  in  that  plane  the  elimination  of  t  gives  a 
relation  between  x  and  y  satisfied  at  each  instant  by  the 
coordinates  of  the   projected  mobile.     They  are,   therefore, 


MOTION   REFERRED  TO  COORDINATE  AXES  87 

the  parametric  equations  of  the  paih  of  the  projected  mobile, 
and  hence  of  the  projected  path. 

Similarly  the  remaining  two  pairs  give  the  projected 
path  in  the  other  two  coordinate  planes.  Two  of  these 
projected  paths  determine  the  path  in  space. 

The  three  equations  of  motion 

x=fit)        y==fi{t)        z=f2{t) 

may  also  be  regarded  as  the  parametric  equations  of  the  path 
in  space. 

88.  Equations  of  the  projected  motion  on  the  coordinate 
axes. — Projecting  the  mobile  on  the  X-axis,  the  projected 
mobile  will  have  the  same  abscissa  as  the  mobile  in  space, 
and  its  equation  of  motion  is  therefore 

x=f{t) 

Similarly  the  remaining  two  equations 

y=fiit)      2=/2(0 

are  the  equations  of  the  projected  motion  on  the  other  two 
axes. 

Hence  each  equation  considered  alone  is  the  equation  of 
the  projected  motion  on  the  corresponding  axis. 

89.  Projections  of  the  velocity. — ^The  velocities  in  the 
projected  motions  on  the  coordinate  axes  are 


f=/'®   !=/''«   1=/^'® 


88  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

But  by  Theorem  III  of  Art.  84  these  are  equal  to  the  pro- 
jections of  the  velocity  of  the  mobile  in  space.     Hence 

V.=^-nt)        F,=|=A'(0        F.=^=/2'(0        [91] 

which  give  the 

Theorem. — The  projections  of  the  velocity  of  a  mobile  on 
the  three  coordinate  axes  equal  the  first  derivatives  of  the  coor- 
dinates of  the  mobile  with  respect  to  the  time. 

go.  Equations  of  motion  of  the  point  which  describes 
the  hodograph. — The  hodograph  is  the  path  described  by 
the  extremity  of  a  vector  equipollent  to  the  velocity,  and 
drawn  from  a  fixed  point,  which  we  may  take  as  0.  The 
coordinates  of  this  extremity  are  at  any  instant 

Hence,  thje  three  eguaiions 

xi=nt)    i/i=/i'(o     2i=/2'(o 

may  be  regarded  either  as  the  equations  of  motion  of  the  point 
which  describes  the  hodograph,  or  as  the  parametric  equations 
of  the  hodograph  itself. 

91.  Projections  of  the  acceleration  on  the  coordinate 
axes. — By  Theorem  IV  of  Art.  84  we  have  at  once 

These  equations  give  the 

Theorem. — The  projections  of  the  acceleration  of  a  mobile 


MOTION  REFERRED  TO  COORDINATE  AXES  89 

on  the  coordinate  axes  equal  the  second  derivatives  of  the  coor- 
dinates of  the  mobile  with  respect  to  the  time. 

92.  R€sum6. — Denoting    the    first    derivatives    of    the 
coordinates  of  a  mobile  with  respect  to  the  time  by 

of        1/        z" 

and  the  second  derivatives  by 

x"        y"        2" 

we  have  the  following  propositions: 

1°  The  velocity  F  of  a  mobile  M  is  the  geometric  sum 
of  the  three  velocities 

:rf        1/        z^ 

drawn  from  M  parallel  to  the  three  axes  of  coordinates, 
whence 

y =?"  +7  +?  [93] 

and  y2=a/2+2/'2+2/2  [94] 

2°  The  acceleration  /  of  a  mobile  M  is  the  geometric 
sum  of  the  three  accelerations 

x"        f        zf' 

drawn  from  M  parallel  to  the  three  coordinate  axes,  whence 

T=¥'  +7'  +z^  [95] 

and  J2=a//2+^y/2+2'/2  [96] 


90 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


3°  In  projecting  these  equipollences  on  a  line  L  whose 
direction  cosines  are  a,  ^,  y,  we  have 


[97] 
[98] 


93.  Most  general  motion  in  which  the  projected  motions 
are  unifonnly  accelerated. — Here 


are  constants,  and  integrating 


J, 


z"=J, 


we  have 


[99] 


Multiplying   these   three   equations   by   the   arbitrary   con- 
stants ^,  ^1  and  ^2  adding, 

Choosing  X,  Xi,  and  X-j  so  that 

Xa'  +  Xih'  +^2^^  =0 


MOTION  REFERRED  TO  COORDINATE  AXES  91 

which  can  always  be  done,  we  have 

kc  +  Aiy  +  XiZ^^Xa+Xih  +  X-ip  [100] 

At  every  instant,  then,  the  coordinates  of  the  mobile 
satisfy  an  equation  of  the  first  degree,  and  hence  the  'path 
lies  in  the  plane  defined  by  equation  [100]. 

Agam,  since  Jx,  Jy,  and  /«  are  constants,  J  is  constant 
both  in  magnitude  and  direction,  and  choosing  this  direction 
as  the  Z-axis  and  the  plane  of  the  path  as  the  ZZ-plane, 
the  equations  of  motion  are  of  the  form 


x=^a-\-a't 
y=0 
z=^c+c't+^Jt^  . 


[101] 


Eliminating  t  between  the  first  and  third  of  these, 

which  is  the  equation  of  the  projected  path  in  the  XZ-plane, 
and  hence  of  the  path  of  the  mobile. 

94.  Motion  in  which  the  projected  motions  are  harmonic. — 
The  equations  of  motion  are 

a; = a  sin  (nt  —  <f)i) 
y  =  b  sin  (nt  —  <f>2) 
z=c  sin  (nt  —  (j)) 

Expanding  sin  {nt~<f>i),  we  see  that  each  of  these  equations 


92  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

is  linear  in  sin  nt  and  cos  nt.     Eliminating  these  two  vari- 
ables, we  have  an  equation 

Ax-\-By  +  Cz+D=Q  [103] 

linear  in  x,  y,  and  z,  which  shows  that  the  path  is  a  plane 
curve. 

Taking  this  plane  as  the  plane  of  XY,  the  equations  of 
motion  take  the  form 


a;=asin  (nf  — ^i)  ' 
y=b  sin  {nt  —  <j)2) 
0  =  0 


[104] 


Solving  the  first  two  of  these  equations  for  sin  ut  and 
cos  ut  and  substituting  in 

cos^nt + sin^nt  =  1 , 

we  obtain  the  equation  of  an  ellipse,  which  is  the  path  of 
the  mobile. 

In  particular,  if  the  difference  of  phase  is  -,  equations 


[104]  take  the  form 

a:  =  asin  {nt- 

-<t>) 

y  =  h  cos  (nt 

-<}>) 

Whence 

x^    y^_ 
a2^62 

1 

The  projections 

of  the  acceleration 

are 

J,=x"=- 

n^x 

Ju-y"=- 

n^y 

MOTION  REFERRED  TO  COORDINATE  AXES  93 

Whence  J^=  —  n'^{x^ +y^)=  —  n^ 

and  Jx=—n^T-  —  =^~J-  — 

r  r 

r  r 

Hence,  the  acceleration  of  the  mobile  is  constantly  directed 
toward  the  centre  of  the  elliptic  path  and  its  algebraic  value  is 

J  =-  —  n^r 

from  which  it  follows  that  it  is  proportional  to  the  distance 
of  the  mobile  from  the  centre. 

If  6  =  a,  the  path  is  cu-cular  and  the  acceleration  is  toward 
the  centre  and  constant.  The  tangential  acceleration  is 
consequently  zero,  and  the  circular  motion  is  uniform. 

95.  Rectilinear  harmonic  motion  considered  as  the  pro- 
jection of  uniform  circular  motion. — Every  harmonic  vibra- 
tion in  a  straight  line  may  be  looked  upon  as  the  projection 
of  a  certain  uniform  circular  motion  on  one  of  its  diameters. 
Suppose  the  harmonic  motion  defined  by 

x  =  a  sin  (nt  —  <f)) 


V 


along  the  line  X'OX  (fig.  31).    From  0  as  centre  with  radius 
a  construct  a  circle,  and  imagine  this  circle  traversed  by  a 


94  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

fictitious  mobile  M  with  constant  angular  velocity  n  and 
in  a  direction  chosen  at  will — that  of  the  arrow,  say. 
A  the  instant  ^  =  0  let  the  fictitious  mobile  M  be  at  C, 


so  that  the  angle  AOC  measured  in  a  direction  opposite 
to  that  of  the  motion,  that  is,  in  the  negative  direction, 

equals  ■^+<f>.    At  any  other  instant  t  let  the  fictitious  mobile 
be  at  M.    Then  the  angle 


0-^AOM 
measured  in  the  positive  direction  is 


'  =  nt-\-4> 


and  the  projection  m  of  the  fictitious  mobile  M  has  the  equa- 
tion of  motion 

x  =  a  cos  d  =  a  cos  [nt—-^  —  ^ 

=  a  sin  {nt  —  4>) 

Hence  m  coincides  with  the  mobile  making  the  harmonic 
vibrations  along  X'OX,  which  vibrations  may  therefore  be 
considered  as  the  projection  of  the  motion  of  M. 


CHAPTER  VIII 

RELATIVE  MOTION.    MOVING  AXES 

96.  Fixed  and  moving  axes. — 97.  Equations  of  absolute  and  rela- 
tive motion. — 98.  Statement  of  the  problem. — 99.  Motion  of 
the  relative  axes. — 100.  Solution  of  the  problem. 

96.  Fixed  and  moving  axes. — Consider  a  mobilfe  M 
describing  a  path  in  space,  and  refer  the  motion  to  a  set  of 
axes  0-XYZ  (fig.  32).  Imagine  a  second  set  of  rectangular 
axes  A-SHZ  in  motion  with  respect  to  0-XYZ. 


Fig.  32. 


The  axes  0~XYZ  are  called  the  fixed  or  absolute  axes, 
and  A-BHZ  the  moving  or  relative  axes.  The  displacement 
r  and  coordinates  x,  y,  z  oi  M  referred  to  the  fixed  axes 


95 


96  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

are  termed  the  absolute  displacement  and  coordinates  of  M, 
while  the  displacement  p  and  coordinates  $,  r),  ^  of  M  referred 
to  the  moving  axes  are  termed  the  relative  displacement 
and  coordinates  of  M.  Finally,  let  I  denote  the  absolute 
displacement  of  A,  the  origin  of  the  moving  axes.  The 
motion  and  path  of  M  referred  to  the  fixed  axes  are  termed 
the  absolute  motion  and  path  of  M,  while  the  motion  and 
path  referred  to  the  moving  axes  are  termed  the  relative 
motion  and  path. 

97.  Equations  of  absolute  and  relative  motion.  —  The 
equations  of  absolute  motion  are 

x-=f{t)         y=fi(t)         z=f2(t)  [105] 

expressing  the  absolute  coordinates  as  functions  of  the 
time,  while  the  equations  of  relative  motion  are 

$=m       v  =  Mt)      C=Mt)         [106J 

expressing  the  relative  coordinates  as  functions  of  the  time. 

98.  Statement  of  the  problem.  —  The  problem  to  be 
considered  is  the  following: 

1°  Given  the  motion  of  the  relative  axes  and  the  relative 
motion  of  the  mobile,  determine  its  absolute  motion. 

2°  Given  the  motion  of  the  relative  axes  and  ihe  absolute 
motion  of  the  mobile,  determine  its  relative  motion. 

99.  Motion  of  the  relative  axes.  —  The  motion  of  the 
relative  axes  is  known  if  at  every  instant  the  position  of 
the  relative  trihedron  A-'B.HZ  is  known,  and  its  position 
is  known  if  the  absolute  coordinates  of  A  and  the  absolute 
direction  cosines  of  the  three  lines  AB,  AH,  and  AZ  are 
given  as  functions  of  the  time. 


RELATIVE  MOTION.    MOVING  AXES 


97 


Denote  by  o,  Oi,  02  the  absolute  coordinates  of  A  and 
let  the  absolute  direction  cosines  of  the  relative  axes  be 
given  by  the  following  schema: 


A 

H 

Z 

X 

a 

b 

c 

Y 

a, 

b, 

c, 

Z 

a. 

/>, 

'-2 

where  0,  oi,  02  and  the  nine  divection  cosines  a  .  .  .  C2  are 
supposed  known  functions  of  the  time. 

100.  Solution  of  the  problem. — First  case. — The  solution 
consists  in  this  case  in  finding  the  initial  absolute  velocity 
and  the  absolute  acceleration  of  M  in  terms  of  the  time. 

From  the  equipollence 


r  =  Z+7)  =  /  +  e  +  Jj  +  C 


projected  on  the  three  axes 


y  =  Oi+ax^  +  hirj+ciC 
2:  =  02  +  a2$  +  62'?  +  C2C 


[107] 


Differentiating  these  equations  we  obtain  the  projections  of 
the  absolute  velocity 


7^  =  (02'  +  as'^  +  62'T;  +  C2'C)  +  (02^'  +  &2lj'  +  C2C')  . 


[108] 


98  ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Denoting  by   Vc  the  vector   whose  projections  on  the 
absolute  axes  are 


[109] 


and  by  Vr  the  vector  whose  projections  on  the  absolute 
axes  are 


[110] 


we  have  for  the  absolute  velocity  of  M  the  vector  Va  given 
by  the  equipollence 

Va=Vc+Vr  [111] 

Now  V^  is  that  component  of  Va  which  is  obtained  by  imagin- 
ing M  invariable  with  respect  to  the  moving  axes,  while 
Vr  is  the  component  obtained  by  imagining  the  relative 
axes  at  rest,  but  the  mobile  to  be  in  motion  with  respect 
to  them.  Vr,  then,  is  the  velocity  which  M  would  appear 
to  possess  to  an  observer  situated  on  the  relative  trihedron 
and  carried  along  with  it,  while  Vc  is  the  velocity  which  a 
mobile  fixed  at  M  and  carried  alone  with  the  relative  trihe- 
dron would  have  due  to  the  motion  of  this  trihedron.  Vc 
is  termed  the  convective  velocity  of  M  and  Vr  the  relaiive 
velocity  of  M. 

The  equipollence  above  then  gives  the 

Theorem. — The  absolute  velocity  of  a  mobile  is  equipollent 
to  the  geometric  sum  of  the  convective  and  relative  velocity 
of  the  mobile. 


RELATIVE  MOTION.    MOVING  AXES  99 

Differentiating  equations  [108] 

y"  =  {o,"  +  a,"^+W'rj  +ci"c)  +  (aif" +6it?"  +ciC") 
+  2(aiT  +  6i'y  +  cir) 

/'  =  (O2"  +a2"e  +  62"!?  +C2"C)  +  (02^"  +62'?"  +C2C") 

+  2(a2'e'+62'7?'  +  C2'c0 


[112] 


Denoting  by  Jc   the  vector  whose  projections   on  the 
absolute  axes  are 

Jc^  =  0i"  +  ai"e+6i"jy+ci"C  [113] 

/c,=02"+a2"e  +  62"TJ+C2"C. 

by  Jr  the  vector  whose  projections  are 

Jr,-a^"  +hi'   +ct:"    ' 

J,^  =  air'  +  6i>?"+ciC"  ■  [114] 

and  by  Jt  the  vector  whose  projections  are 

Jt^  =  2{a'^'  +6'y  +c'c')    ■ 

J<^  =  2(ai'f'+6i')j'+ci'c')   •  [115] 

J.,  =  2(a2'e'  +  62'V  +  C2'cO  . 

we  have  for  the  absolute  acceleration  of  M  the  vector  Ja 
given  by  the  equipoUence 


Ja—Jc'^Jr'^  J  t 


[116] 


100         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Now  Jc  is  the  component  of  the  absolute  acceleration 
due  to  the  motion  of  the  axes,  that  is,  it  is  what  the  abso- 
lute acceleration  would  become  if  the  mobile  were  invari- 
able with  respect  to  the  relative  axes,  and  Jr  is  the  com- 
ponent due  to  the  relative  motion,  that  is,  it  is  what  the 
absolute  acceleration  would  become  if  the  relative  axes 
were  at  rest,  and  therefore  Jc  is  the  convective  and  Jr  the 
relative  acceleration  of  the  mobile.  The  third  component 
is  termed  the  turning  acceleration.  The  equipollence  above, 
then,  gives  the 

Theorem. — The  absolute  acceleration  of  a  mobile  is  equi- 
pollent to  the  geometric  sum  of  the  convective  acceleration, 
the  relative  acceleration,  and  the  turning  acceleration. 

Remark. — The  turning  acceleration  depends  only  on 
the  velocities  of  the  relative  axes  and  the  mobile,  and  hence 
vanishes  when  the  relative  axes  are  at  rest,  and  when  the 
mobile  is  relatively  at  rest,  but  both  the  convective  and 
the  relative  acceleration  may  vanish  and  still  leave  the 
turning  acceleration  not  zero.  It  will,  however,  then  be 
constant. 

Second  case. — In  this  case  the  initial  relative  velocity 
and  the  relative  acceleration  as  a  function  of  the  time  is 
required.  These  are  at  once  obtained  by  solving  equa- 
tions [111]  and  [116]  for  F^  and  Jr,  giving 

Vr  =  Va-Vc  i  [117] 

Jr-Ja-Jc-Jt  [118] 

the  components  of  the  relative  velocity  required  in  calcu- 
lating Jt  bemg  obtained  by  projecting  [117]  on  the  relative 
axes. 


CHAPTER  IX 

MECHANICS  OF  A  FREE  PARTICLE 

101.  Material  point  or  particle. — 102.  Purpose  of  this  book. — 
103.  The  problem  of  mechanics. — 104.  The  role  of  observation 
and  experiment. — 105.  Principles  of  mechanics. — 106.  The  abso- 
lute axes. — Isolated  particle. — 107.  First  principle  of  mechan- 
ics.— 108.  Meaning  of  the  first  principle. — 109.  Field  of  force. — 
110.  Uniform  field  of  force. — 111.  Constant  field  of  force. — 
112.  Superposition  of  fields  of  force. — 113.  Second  principle 
of  mechanics. — 114.  Meaning  of  the  second  principle. — 115.  Reac- 
tion of  the  particle  on  the  field. — 11&.  Third  principle  of  mechan- 
ics.— 117.  Cleaning  of  the  third  principle. — 118.  Properties  of 
the  coefficients  Pij, — 119.  Definition  of  mass. — 120.  Force. — 
121.  Observations  on  the  notion  of  force. — 122.  Composition 
of  forces.  Resultant. — 123.  Decomposition  of  forces.  Compo- 
nents.— 124.  Equilibrium  of  a  free  particle. — 125.  Force  acting 
on  a  particle  at  rest. — 126.  New  definition  of  direction  of  force. — 
127.  Tangential  and  normal  force. — 128.  Resistance. 

loi.  Material  point  or  particle. — In  the  preceding  chap- 
ters we  have  studied  the  motion  of  a  geometric  'point,  and 
have  seen  that  for  the  complete  determination  of  this  motion 
only  the  notions  of  space  antl  time  are  necessary.  Knowing 
at  any  instant  the  displacement  and  velocity  of  the  point 
and  knowing  the  acceleration  as  a  function  of  the  time  we 
can  assign  the  motion  completely.  To  do  this  it  is  only 
necessary  to  integrate  the  differential  equations  of  motion 
of  the  point 

101 


102         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

y"=Jy  •  [119] 

under  the  restrictions  that  the  displacement  and  velocity 
shall  have  the  assigned  values  at  the  given  instant,  which 
restrictions  serve  to  determine  the  constants  of  integration. 
To  effect  the  integration  and  obtain  the  equations  of  motion 
in  their  finite  form  is  a  problem  of  pure  mathematics,  which, 
while  theoretically  possible,  may  be  of  great  practical  difh- 
culty.  However,  with  it  as  such  we  have  not  to  do,  the 
problem,  considered  as  a  problem  of  kinematics,  being 
solved  when  we  have  obtained  the  differential  equations  of 
the  motion. 

As  soon,  however,  as  we  begin  to  apply  the  theorems 
of  kinematics  to  the  material  universe  we  meet  the  difficulty 
that  the  geometric  point  has  no  existence  in  the  physical 
world,  and  in  order  to  discuss  the  motion  of  bodies  as  they 
exist  we  must  make  certain  assumptions  concerning  them. 

We  therefore  place  ourselves  under  so-called  ideal  con- 
ditions and  assume  a  portion  of  matter  which  we  conceive 
as  concentrated  at  a  geometric  point.  It  is  with  this  material 
point  or  'particle  that  we  deal  in  theory,  and  in  applications 
we  neglect  the  dimensions  of  the  actual  portions  of  matter 
under  consideration,  thereby  obtaining  a  more  or  less  close 
approximation  to  the  true  motion.  Certain  extended  bodies, 
spheres  for  example,  behave  under  certain  conditions  exactly 
as  a  material  point  situated  at  their  centre,  so  that  we  obtain 
their  true  motions. 

102.  Purpose  of  this  book. — It  is  my  purpose  here  to 
develop  in  a  rigorous  manner  the  elementary  theory  of  the 
motion  of  a  material  point  or  particle,  s^nd  to  thereby  furnish 
a  point  of  departure  for  the  consideration  of  the  motion  of 


MECHANICS  OF  A  FREE  PARTICLE  103 

bodies  as  they  actually  occur  in  the  material  universe.  This 
elementary  exposition  will  then  serve  as  an  introduction 
to  that  portion  of  the  subject  which  is  known  as  rational 
mechanics,  and  in  which  the  mathematical  theory  of  the 
motion  of  portions  of  matter  of  ideal  forms  is  investigated 
under  ideal  conditions,  leaving  the  special  applications  to 
the  particular  sciences. 

103.  The  problem  of  mechanics. — Since,  as  we  have  seen, 
the  motion  of  a  geometric  point  is  known  as  soon  as  we 
know  its  initial  displacement  and  velocity  and  its  acceleration 
as  a  function  of  the  time,  so  the  motion  of  a  particle  is  known 
as  soon  as  we-  can  assign  the  displacement  and  velocity  at 
a  varticidar  instant  and  the  acceleration  at  every  instant. 

In  kinematics  the  acceleration  was  given  as  one  of  the 
known  quantities  of  the  problem,  but  in  mechanics,  on  the 
other  hand,  the  particle  and  the  physical  conditions  under 
which  the  motion  takes  place  are  given,  and  the  problem  is: 

To  determine  the  law  of  the  acceleration,  and  hence  the 
motion. 

The  converse  problem  may  also  be  proposed: 

Determine  what  physical  conditions  will  produce  a  required 
motion. 

We  have  here  an  essentially  different  and  very  much 
larger  problem  than  was  discussied  in  kinematics.  There 
we  could  assign  to  the  mobile  any  displacement,  velocity, 
and  acceleration  we  pleased  and  determine  what  motion 
would  take  place,  or  we  could  imagine  the  mobile  to  have 
any  motion  we  pleased  and  investigate  the  manner  in  which 
the  displacement,  velocity,  and  acceleration  varied  along 
the  path.  In  mechanics,  on  the  contrary,  we  find  the  par- 
ticle under  certain  physical  conditions,  and  its  subsequent 
motion  is  entirely  independent  of  us;  we  can  only  investi- 
gate the  manner  in  which  it  actually  takes  place. 


104         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

104.  The  role  of  observation  and  experiment. — For  this 
we  must  have  recourse  to  observation  and  experiment. 
We  can  to  some  extent  control  the  conditions  under  which 
a  particle  shall  move,  but  for  the  most  part  we  are  reduced 
to  observation  both  on  the  conditions  and  the  resulting 
motions.  The  motion  of  bodies  on  the  Earth  furnishes 
some  data,  but  by  far  the  most  important  knowledge  is 
o])tained  by  observing  the  motions  of  the  bodies  of  the 
solar  system  about  each  other. 

105.  Principles  of  mechanics. — The  results  of  these  obser- 
vations have  been  reduced  to  three  independent  statements, 
known  as  the.  principles  of  mechanics. 

While  not  directly  verifiable,  they  accord  with  experi- 
ence to  such  a  degree  that  departures  cannot  be  detected, 
and  the  positions  of  the  planets  and  their  satellites  are  in 
such  close  agreement  with  their  predicted  places  that  this 
agreement  is  of  the  nature  of  a  demonstration. 

106.  The  absolute  axes. — In  kinematics  the  choice  of 
the  absolute  axes  was  arbitrary.  The  state  of  affairs  in 
mechanics  is  different.  The  principles  just  spoken  of  are 
asserted  true  of  the  motion  of  a  particle  referred  to  a  par- 
ticular set  of  axes  invariably  connected  with  the  so-called  fixed 
stars.  These  I  term  the  absolute  axes.  Referred  to  any 
other  set  the  principles  must  be  modified,  and  this  necessary 
modification  will  be  studied  in  a  subsequent  chapter. 

It  will  then  be  assumed,  and  this  is  fundamental  in  the 
theory,  that  unless  otherwise  stated  the  absolute  axes  are  under 
consideration. 

Isolated  particle. — If  we  conceive  all  the  matter  of  the 
universe  except  one  particle  to  cease  to  exist,  this  particle 
is  termed  isolated.  Of  course  no  particle  ever  is  isolated, 
but  we  may  assert  a  statement  true  of  an  isolated  particle, 
meaning  thereby  that  it  describes  what  would  happen  were 


MECHANICS  OF  A  FREE  PARTICLE  105 

we  under  these  ideal  conditions.  A  set  of  geometric  points 
may  be  imagined  as  taking  the  place  of  the  fixed  stars  and 
thus, determining  the  absolute  axes,  they  being  a  geometric 
and  not  a  physical  concept. 

107.  First  principle  of  mechanics. — The  first  principle 
may  be  stated  as  follows: 

An  isolated  particle  has  no  acceleration  with  respect  to 
the  absolute  axes. 

This  principle  is  also  termed  the  principle  of  inertia. 

108.  Meaning  of  the  first  principle. — Since  an  isolated 
particle  has  no  acceleration  its  velocity  is  constant  and 
hence  it  either  remains  at  rest  or  describes  a  straight  line 
with  constant  speed;  that  is,  its  motion  is  rectilinear  and 
uniform. 

In  other  words,  an  isolated  particle  persists  in  its  state 
of  motion,  and  this  property  is  termed  inertia. 

Remark. — In  virtue  of  this  principle,  a  particle  acceler- 
ated with  respect  to  the  absolute  axes,  that  is,  a  particle 
possessing  an  absolute  acceleration,  must  be  in  the  presence 
of  other  particles. 

It  is  to  be  observed  that  no  assertion  is  made  concern- 
ing a  particle  in  the  presence  of  other  particles — it  may  or 
may  not  be  accelerated.  What  is  asserted  is  this:  //  the 
particle  is  accelerated,  then  it  is  in  the  presence  of  other  par- 
ticles. 

109.  Field  of  force. — A  portion  of  space  within  a  closed 
surface — which  may  be  entirely  at  infinity — is  termed  a 
field  of  force  when  a  particle  abandoned  at  rest  at  any  point 
in  it  takes  an  accelerated  motion.  The  space  occupied 
by  the  solar  system  and  bounded  by  a  spherical  surface 
through  the  nearest  fixed  star  is  an  example  of  such  a  field. 

There  is  good  reason  to  believe  that  this  field  extends 
through  the  space  occupied  by  the  stars  themselves,  and 


106         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

we  are  accustomed  to  regard  infinite  space  as   a   field   of 
force. 

no.  Uniform  field  of  force. — If  the  acceleration  which 
the  particle  takes  is  independent  of  the  point  where  it  is 
abandoned,  and  is  the  same  for  all  points  of  the  field  at 
the  same  instant,  the  field  is  termed  uniform. 

111.  Constant  field  of  force. — If  in  a  uniform  field  of 
force  the  acceleration  is  independent  of  the  time  the  field 
is  termed  constant. 

112.  Superposition  of  fields  of  force. — By  the  first  prin- 
ciple, wherever  there  is  a  field  of  force  there  must  be  matter. 
Suppose  the  matter  present  to  consist  of  n  particles  any- 
how distributed.  Let  n  =  p4g.  Then  p  of  those  particles 
imagined  as  present  alone  would  produce  a  field  (Pi,  and 
the  remaining  q  imagined  as  present  alone  would  produce 
a  field  ^2.  If  all  n  are  present  we  may  imagine  the  two 
fields  ^1  and  02  as  supposed  producing  a  resultant  field  <P, 
of  which  (Pi  and  (P2  are  termed  components.  The  field  0i 
alone  would  produce  a  certain  acceleration  Ji  in  a  particle  M 
placed  in  it.  Similarly  </>2  alone  would  produce  a,  in  general 
different,  acceleration  J2  on  the  same  particle  at  the  same 
point.  The  resultant  field  0  produces  an  acceleration  ,/ 
on  this  particle.     How  are  these  three  accelerations  related? 

113.  Second  principle  of  mechanics. — The  question  is 
answered  by  the  following  principle,  known  as  the  second 
principle  of  mechanics: 

The  acceleration  which  a  particle  takes  in  a  resultant  field 
of  force  is  the  geometric  sum  of  the  accelerations  produced  by 
the  component  fields,  and  is  independent  of  the  particle  and  of 
its  motion. 

114.  Meaning  of  the  second  principle. — From  this  prin- 
ciple we  draw  the  following  conclusions: 

1°  The  acceleration  at  any  point  in  a  field  of  force  is 


MECHANICS  OF  A  FREE  PARTICLE  107 

the  same  whether  the  particle  be  abandoned  there  at  rest 
or  pass  the  point  with  a  velocity  different  from  zero. 

2°  The  acceleration  at  any  point  in  the  field  is  the  same 
for  all  particles. 

3°  If  J  be  the  acceleration  produced  by  the  resultant 
field  0  and  J\  and  J2  the  accelerations  produced  by  the 
component  fields  (?i  and  <p2,  these  three  accelerations  satisfy 
the  equipollence 

J=Ji+T2  [120] 

4°  Since  the  same  reasoning  can  be  applied  to  any  number 
of  component  fields  ^i  . .  .  0n,  we  have  in  general 


J=Ji +...+/„  [121] 

115.  Reaction  of  the  particle  on  the  field. — Conceive  a  par- 
ticle M  in  a  field  (/>  produced  by  n  other  particles  Mi  .  . .  M„. 
Now  the  particle  M  together  with  any  (n  —  1)  of  the  n  particles 
may  be  looked  upon  as  producing  a  field  which  acts  on  the 
remaining  one.  That  is,  conceive  Mi  (say)  in  the  field 
produced  by  M,  M2  •  •  •  Mn-  This  particle  Mi  takes  a 
certain  acceleration  Ji  at  the  same  instant  that  M,  looked 
upon  as  acted  on  by  the  field  of  Mi  . .  .  M„,  takes  the  accelera- 
tion J.    Is  there  any  relation  between  J  and  Ji  ? 

This  question  is  answered  by  the  following  principle. 

116.  Third  principle  of  mechanics. — ^This  principle  may 
be  stated  as  follows: 

'  Two  isolated  particles  under  their  mutual  actions  take  acceler- 
ations in  opposite  directions  along  the  line  joining  them,  and 
these  accelerations  are  in  a  constant  ratio. 


108         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

117.  Meaning  of  the  third  principle. — From  this  principle 
we  conclude  that  two  particles  Mi  and  lHj  take  accelera- 
tions Jij  and  Jji  which  satisfy  the  equipollence 

j7i  +  PHJir=0  [122] 

and  the  algebraic  equation 

Ji]=P]iJj%  [123] 

where  pji  is  a  constant  depending  on  the  two  particles. 

118.  Properties  of  the  coefficients  p^j. — Consider  any 
three  particles  Mi,  M,-,  and  Mk.  In  the  field  of  Mi  the 
two  particles  Mj  and  Mk  take  accelerations  /,-,  and  Jki 
which  satisfy 

'^  i)  "^  Pji'^  a 

Jik  =  PkiJki 

Whence 

"  ij   Pji 

Jik       Pki 

since  by  the  second  principle 

J  ji  —  Jki 

Similarly,  considering  the  fields  produced  by  Mj  and  Mk, 
we  have 

Jik_pki 
Jit       Pki 

and 

Jki      Pik 
Jki      Pik 


MECHANICS  OF  A  FREE  PARTICLE  109 

Multiplying  these  equations  we  have 

Jji^   Jjk^    Jh  ^  Pji      Pki_    Prk 
J  lie   Jji     J  kj       Pkt     Pi)       Pjk 

and  making  use  of  the  relations 

"  tj  ^^  PH"  I'i 
Jjk  =  PkjJkj 
Jki  =^  PihJik 


we  obtain 


1- 


puiPaPik 


Whence  p.j^  =  P^-^  [124] 

pki    Pa 

iiQ.  Definition    of   mass. — Writing    the    coefficients    pa 
m  the  form 


the  equipoUence 

Jii+P]iJii=0 

becomes 

miJij  +  mjJ,i=i 

or 

'^iJii  —  T^jJii 

[125] 


and  relation  [124]  becomes 

rrij  _mi  /nii    rrii    mj_ 


TTik    rriki  nij-    mk   w» 


110         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Hence,  the  coefficients  rrii,  rrij,  rrik  .  .  .  rrir  are  absolute  constants 
depending  only  on  the  particles  themselves  and  not  on  their 
accelerations. 

These  constants  are  termed  the  masses  of  the  particles, 
and  since  only  their  ratios  are  determined,  one  of  them  can 
be  arbitrarily  assigned,  and  the  rest  then  become  known. 

120.  Force. — The  word  force  does  not  enter  into  the 
principles  of  mechanics,  as  they  have  been  stated,  and  the 
problems  of  mechanics  can  be  solved,  in  so  far  as  they  are 
capable  of  solution,  without  ever  making  use  of  the  term. 

But  it  is  often  convenient  to  employ  the  term  in  abbrevi- 
ating statements,  and  it  is  introduced  as  follows: 

When  a  particle  M  of  mass  m  is  accelerated  by  being  in 
the  presence  of  other  particles  it  is  said  to  be  subjected  to 
a  force  F  defined  by  the  equipoUence  (fig.  33) 

F  =  nJ  [126] 

The  force  acting  on  a  particle 
"^  is,  then,  a  vector  whose  direction 
is  that  of  the  acceleration  and 
whose  magnitude  is  the  product  of  the  mass  by  the  ac- 
celeration. 

121.  Observations  on  the  notion  of  force. — Force  is  thus 
seen  to  be  a  purely  mathematical  and  not  a  physical  concept. 
It  has  no  objective  existence,  but  is  merely  a  name  for  a  certain 
product  in  mechanics.  To  say  that  a  force  acts  on  a  particle 
is  simply  to  assert  that  the  particle  is  accelerated,  and  this 
acceleration  is  all  that  is  ever  observable.  We  may  produce 
the  sensation  of  effort  by  attempting  to  move  bodies,  but  in 
every  case  the  force  which  we  exert  is  only  a  component  of 
the  acting  force,  and  inasmuch  as  a  particle  at  rest  may  be 
conceived  as  having  accelerations  whose  geometric  sum  is 


Fig.  33. 


MECHANICS  OF  A  FREE  PARTICLE  HI 

zero,  so  it  may  be  conceived  as  acted  on  by  forces  one  of 
which  is  exerted  by  ourselves. 

122.  Composition  of  forces.  Resultant. — The  second  prin- 
ciple gives  at  once  the  rule  for  the  composition  of  forces.  For 
if  a  system  of  forces 

Fi...Fn 

act  on  a  particle  M  of  mass  m,  producing  accelerations 

J\...Jn 

then,  by  the  second  principle,  the  resultant  acceleration  is 

/=  JT+  . . .  +Z» 
and  the  resultant  force  is 

F=mJ=m(j[+  .  .  .  +JJ 
or  F  =  ¥i+  ..,+Fn  [127] 

Hence,  the  resultant  of  a  system  of  forces  acting  on  a 
'particle  is  their  geometric  sum. 

From  this  follows  that: 

1°  The  projection  on  a  plane  of  the  resultant  of  a  sys- 
tem of  forces  acting  on  a  particle  coincides  with  the  result- 
ant of  the  projections  of  the  forces  on  that  plane. 

2°  The  projection  on  an  axis  of  the  resultant  of  a  sys- 
tem of  forces  acting  on  a  particle  coincides  with  the  alge- 
braic sum  of  the  projections  of  the  forces  on  that  axis. 

3°  The  resultant  of  two  forces  is  the  diagonal  of  the 


112         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

parallelogram  constructed  on  them.  This  is  termed  their 
parallelograjn. 

4°  The  resultant  of  three  forces  is  the  diagonal  of  the 
parallelopiped  constructed  on  them.  This  is  termed  their 
parallelopiped. 

5°  The  resultant  of  a  system  of  forces  is  equipollent  to 
the  vector  which  closes  any  one  of  the  polygons  formed  by 
placing  them  end  to  end.  This  polygon  is  termed  the  poly- 
gon of  the  forces. 

6°  In  particular,  the  polygon  of  two  forces  becomes  a 
triangle  and  is  termed  their  triangle. 

123.  Decomposition  of  forces.  Components. — To  decom- 
pose a  force  is  to  replace  it  by  a  system  of  forces  whose 
resultant  is  equipollent  to  the  given  force.  The  forces  of 
the  system  are  termed  components  of  the  given  force. 

In  particular  a  force  may  be  decomposed  in  a  single 
manner  into: 

1°  Two  forces  in  two  directions  arbitrarily  chosen  in 
any  plane  through  the  force  itself. 

For  in  any  plane  through  the  force,  one  and  but  one 
parallelogram  can  be  constructed  having  its  sides  in  assigned 
directions  and  having  a  given  diagonal. 

2°  Two  others,  one  of  which  is  chosen  at  will. 

For,  in  the  plane  defined  by  the  given  force  and  the 
arbitrarily  chosen  force,  one  and  but  one  parallelogram 
can  be  constructed  having  an  assigned  side  and  diagonal. 

3°  Two  others,  of  which  one  has  an  arbitrary  direction 
and  the  other  is  in  a  plane  of  arbitrary  orientation.  In 
particular,  into  two  others  of  which  one  has  an  arbitrary 
direction  and  the  other  is  in  a  plane  perpendicular  to  this 
direction. 

For,  in  the  plane  defined  by  the  arbitrarily  chosen  direc- 
tion and  the  given  force,  the  second  direction  is  determined 


MECHANICS  OF  A  FREE  PARTICLE  113 

by  its  intersection  with  the  arbitrarily  chosen  plane,  and 
the  problem  is  reduced  to  the  first  case. 

In  the  particular  case  where  the  arbitrarilj-  chosen  plane 
is  perpendicular  to  the  chosen  direction,  the  components  are 
the  projections  of  the  given  force  on  the  chosen  direction 
and  the  normal  plane. 

4°  Three  others  along  directions  not  in  the  same  plane 
but  otherwise  arbitrary. 

For  one,  and  but  one  parallelopiped  can  be  constructed 
having  an  assigned  diagonal  and  assigned  directions  for  its 
edges. 

In  particular,  the  components  maj'  be  taken  parallel  to 
three  coordinate  axes,  and  are  then  equipollent  to  the  pro- 
jections of  the  given  forces  on  these  axes. 

124.  Equilibrium  of  a  free  particle. — When  the  resultant 
acceleration  which  a  system  of  forces  imposes  on  a  particle 
is  zero,  the  particle  is  said  to  be  in  equilihrium. 

The  system  of  forces  is  also  said  to  be  in  equilihriym. 

Since  the  resultant  force  and  the  resultant  acceleration 
are  connected  by  the  equipollence 

F  =  ml 

it  follows  that  if  J  is  zero  F  is  also  zero. 

Hexce,  the  necessary  and  sufficient  condition  that  a  sys- 
tem of  forces  acting  on  a  particle  he  in  equilihrium  is  that 
their  resultant  shall  he  zero. 

According  as  one  wishes  to  study  the  conditions  of  equilib- 
rium by  geometric  or  analytic  methods,  one  employs  the 
polygon  of  forces  or  their  projections  on  the  coordinate 
axes. 

Corollaries. — 1°  In  order  that  two  forces  acting  on 
a  particle  be  in  equilibrium  it  is  necessary  and  sufficient 
that  they  be  equal  and  opposite. 


114         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

T  In  order  that  three  forces  acting  on  a  particle  be  in 
equiHbrium  it  is  necessary  and  sufficient  that  they  be  equi- 
pollent to  the  three  sides  of  a  triangle  taken  in  the  same 
order. 

3°  If  a  system  of  forces  acting  on  a  particle  is  in  equilib- 
rium, each  of  them  is  equal  and  opposite  to  the  resultant  of 
the  others. 

4°  If  a  system  of  forces  is  in  equilibrium,  the  system 
formed  by  theu-  projections  on  any  plane  or  axis  is  in 
equilibrium. 

5°  If  a  system  of  forces  in  equilibrium  act  on  a  particle 
in  motion,  its  motion  is  uniform  rectilinear,  as  if  these  forces 
did  not  exist. 

6°  More  generally,  whatever  be  the  forces  acting  on  a 
free  particle,  a  system  of  forces  in  equilibrium  may  be  intro- 
duced or  suppressed.  For  every  system  of  forces  may  be 
replaced  by  its  resultant,  and  this  is  not  changed  by  the 
addition  or  subtraction  of  forces  whose  geometric  sum  is 
zero. 

125.  Force  acting  on  a  particle  at  rest. 

Theorem. — Whenever  the  velocity  of  a-  'particle  is  zero, 
and  therefore  whenever  the  direction  of  motion  changes,  and  in 
particular  when  the  particle  starts  from  rest,  the  force  acting 
is  tangent  to  the  path. 

and  if  7  =  0,  then  the  direction  of  this  limit  is  the  direction 

toward  which  Vi  tends  and  is  therefore  tangent  to  the  path. 

Hence  F  {=mJ)  is  also  tangent  to  the  path  at  this  point. 

126.  New  definition  of  direction  of  force. — We  may  then 
dehne  the  direction  of  the  force  as  the  direction  of  the  first 


MECHANICS  OF  A  FREE  PARTICLE  .115 

element  of  the  path  which  the  particle  describes  starting  from 
rest. 

127.  Tangential  and  normal  force. — In  general  the  force 
is  not  tangent  to  the  path,  but  it  always  lies  in  the  oscu- 
lating plane,  since  the  acceleration  is  in  this  plane.  It 
may  therefore  be  decomposed  into  two  components,  one 
along  the  tangent  and  the  other  along  the  principal  normal. 

Projecting  the  equipoUence 

F=mJ 
on  the  tangent,  we  liave 

"dt^ 


(Ps 
F,==mJs==m-TT2  [128] 


and  on  the  normal 


^"-""J'^jifi)'  [129] 


P 

128.  Resistance. — When  a  force  produces  a  negative 
acceleration  it  is  termed  a  resistance,  and  this  is  the  case 
when  the  direction  of  the  acceleration  makes  an  obtuse 
angle  with  the  direction  of  the  velocity. 


CHAPTER  X 

THE  UNITS  OF  MECHANICS 

129.  The  units  of  kinematics. — 130.  The  units  of  mechanics. — 
131.  The  three  fundamental  units.  Derived  units. — 132.  Sys- 
tems of  units.— 133.  C.G.S.  System.— 134.  M.K.S.  System.— 
135.  English  System. — 136.  Units  of  astronomy. — 137.  Units 
of  force. — 138.  Remark. 

129.  The  units  of  kinematics. — In  geometry  a  single 
unit  is  sufficient;  that  of  length  and  any  one  of  the  quanti- 
ties, centimeter,  meter,  kilometer,  inch,  foot,  yard,  mile, 
etc.,  can  equally  well  be  used.  In  kinematics  a  new  ele- 
ment was  introduced,  namely,  time,  and  the  unit  adopted 
was  the  mean  solar  second.  It  is  understood  that  when 
more  convenient  any  multiple  of  this,  as  a  minute,  hour, 
day,  year,  etc.,  may  be  used. 

130.  The  units  of  mechanics.  -In  mechanics  it  was 
necessary  to  introduce  a  second  new  quantity,  namely, 
mass.  The  gram  or  any  multiple  or  submultiple  may  be 
taken  as  the  unit  of  mass. 

131.  The  three  fundamental  units.  Derived  units. — 
As  we  shall  see,  these  three  units  of  length,  time,  and  mass 
being  defined,  all  the  rest  are  determined  as  they  are  expres- 
sible in  terms  of  these.  These  three  are  accordingly  termed 
fundamental  units  and  the  rest  are  known  as  derived  units. 

116 


THE  UNITS  OF  MECHANICS  117 

132.  Systems  of  units. — Various  systems  of  units  may 
be  built  up  depending  on  the  choice  of  the  fundamental 
units.    Three  are  in  common  use,  as  follows: 

133.  C.G.S.  System. — If  the  centimeter,  gram,  and  second 
are  taken  as  fundamental  imits,  the  system  is  termed  the 
Centimeter-Gram-Second  System. 

134.  M.K.S.  System. — If  the  meter,  kilogram,  and  sec- 
ond are  taken  as  fundamental  units,  the  system  is  termed 
the  Meter-Kilogram-Second  System. 

135.  English  System. — If  the  foot,  pound,  and  second  are 
taken  as  fundamental  units,  the  system  is  termed  the  Foot- 
Pound-Second  or  English  System. 

136.  Units  of  astronomy. — Because  of  the  large  mag- 
nitudes there  dealt  with  it  has  been  found  convenient  in 
astronomy  to  adopt  as  the  unit  of  distance  the  mean  dis- 
tance from  the  Earth  to  the  Sun,  and  as  the  unit  of  mass 
the  mass  of  the  Earth  or  Sun.  The  unit  of  time  is  the  second 
day  or  year.    These  are  the  astronomical  units. 

137.  Units  of  force. — The  unit  of  force  being  a  derived 
unit  depends  on  the  fundamental  units,  and  hence  is  differ- 
ent in  the  three  systems  described  above. 

From 

F=mJ 

we  see  that  F  =  l  when  m  =  l  and  J  =  l. 

Hence,  the  unit  of  force  is  that  force  which  acting  on  unit 
m,a£s  for  unit  time  gives  it  unit  acceleration. 

In  the  C.G.S.  system  it  is  called  the  dyne,  in  the  M.K.S. 
system  it  has  no  name,  but  is  referred  to  as  the  M.K.S,  unit 
of  force,  while  in  the  English  system  it  is  known  as  the 
poundal. 

The  remaining  derived  units  will  be  defined  as  occasion 
arises. 


118         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

138.  Remark. — It  is  to  be  carefully  observed  that  just 
as  soon  as  the  unit  of  length  is  chosen  those  of  area  and 
volume  are  fixed,  and  as  soon  as  the  units  of  length  and 
time  are  chosen  those  of  velocity  and  acceleration  are  fixed ; 
similarly,  as  soon  as  the  units  of  length,  time,  and  mass 
are  chosen  all  the  derived  units  of  mechanics  are  deter- 
mined, although  not  all  of  them  have  names. 


CHAPTER  XI 

GENERAL  NOTIONS  ON  THE  FORCES  OF  NATURE 

139.  Principle  of  the  equality  of  action  and  reaction. — 140.  Analytic 
expression  of  the  common  force  between  two  free  particles. — 
141.  Central  forces. — 142.  Universal  gravitation. — 143.  Note. 

139.  Principle  of  the  equality  of  action  and  reaction. — 
Newton  enunciated  the  following  principle  under  the  name 
of  the  Principle  of  the  Equality  of  Action  and  Reaction. 

If  a  particle  M  is  acted  on  by  a  force  F  due  to  a  second  par- 
ticle M',  then  this  force  is  directed  along  the  line  MM'  and  M' 
is  ojcted  on  by  a  force  F'  equal  and  opposite  to  F. 

This  is  nothing  more  than  a  restatement  of  the  third  prin- 
ciple in  terms  of  force,  for  if  M  of  mass  m  is  acted  on  by  a 
force  F,  then  it  takes  an  acceleration  J  defined  by 

m 

and   by  the   third   principle   M'   takes   an   acceleration  J' 

defined  by 

7,          -J        m  J.         F 
J'=-p.J= J=-  ^ 

m  m 

and  hence  is  acted  on  by  a  force  F'  defined  by 

F'^m'r=-F 

119 


120         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

The  forces  F  and  F'  are  then  equal  in  magnitude,  and  are 
either  directed  toward  each  other,  as  in  fig.  34i,  or  away  from 
each  other,  as  in  fig.  342. 

One  of  these  forces,  either  F  or  F',  is  termed  the  action, 
and  the  other  the  reaction. 

By  the  second  principle  of  mechanics  this  principle  of  the 
equality  of  action  and  reaction  is  immediately  extended  to 
the  mutual  actions  of  two  systems  of  particles  (*S)  and  {S'). 


Pig.  34i. 


Fig.  34,. 


//  the  particles  of  a  system  (S)  exert  certain  forces  on  the 
particles  of  a  system  (S'),  then  the  particles  of  (S')  exert  on 
the  particles  of  (S)  equal  and  opposite  forces. 

140.  Analytic  expression  of  the  common  force  between 
two  free  particles. — Denoting  the  two  particles  by  M  and 
M',  and  their  masses,  velocities,  and  accelerations  by  m,  V, 
J  and  m',  V,  J',  we  have 

F  =  mJ  =  m'J' 


Now  by  the  second  principle  J  does  not  depend  on  m 
or  V,  and  J'  does  not  depend  on  m'  or  V.  Hence  /  can 
be  a  function  of  m' ,  V ,  and  r  only,  where  r  is  the  distance 
between  M  and  M' .    Similarly  J  J'  can  be  a  function  of  m, 


GENERAL  NOTIONS  ON  THE  FORCES  OF  NATURE.     121 

V,  and  r  only.     Again,  since  M  and  M'  may  be  interchanged 
without  affecting  J  and  J'  in  magnitude,  the  law  of  depend- 
ence must  be  the  same  in  the  two  cases. 
We  may  therefore  write 

J  =  <f>(m',V',r)        J'  =  4>{m,  V,r) 

Whence  'm-<f){m',  V,  r)=m' -(fiim,  V,  r) 

<f>im',V',r)_cf>im,  V,r) 


Therefore 


m  m 


Now,  since  m,  m!,  V,  V  are  all  independent  the  two 
members  of  this  equation  can  depend  on  r  only,  and  we 
may  write 

m'  m 

Whence  F  =  mJ  =  m'J'  =  mm'  (P  (r)  [130] 

141.  Central  forces. — A  force  of  the  type 

F  =  m-7n'-0(r) 

is  termed  a  central  force,  and 

Hexce,  All  forces  between  free  particles  are  central  forces. 

142.  Universal  gravitation. — Newton,  interpreting  Kep- 
ler's laws  of  the  motions  of  the  planets,  showed  that  all 
the  motions  of  the  celestial  bodies  are  determined  with  a 
remarkable  precision  if  we  assume  that: 

The  central  force  between  two  free  'particles  is  one  of  attrac- 
tion and  varies  inversely  as  the  square  of  their  distance  apart. 
In  this  case 


122         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

where  G  is  a  constant  for  any  two  particles.  A  discussion 
of  Kepler's  laws  also  shows  that  G  is  the  same  for  every  pair 
oj  particles.    In  the  C.G.S.  system  of  units 

G  =  ^  (nearly) 

The  central  force  of  attraction  of  any  two  free  particles 
may  then  be  written 

F-g"^  [131] 


and  for  this  reason  central  forces  of  this  type  are  often 
called  Newtonian  forces. 

143.  Note. — I  do  not  here  include  electric,  magnetic, 
and  electromagnetic  forces  between  free  particles,  for  the 
reason  that  these  belong  rather  to  theoretical  physics  than 
to  rational  mechanics.  Forces  of  capillarity  and  cohesion 
are  not  included  for  the  reason  that  the  particles  are  not 
there  free. 

As  to  the  force  between  two  free  particles  when  the 
distance  between  them  is  small,  that  is,  comparable  to 
molecular  distances,  and  which  does  not  seem  to  be  New- 
tonian, special  monographs  should  be  consulted. 


CHAPTER  XII 

DETERMINATION  OF  THE  LAW  OF  FORCE  FROM  THE 
MOTION  PRODUCED 

144.  Finite  and  differential  equations  of  motion. — 145.  The  two 
general  problems  of  the  mechanics  of  a  particle  in  terms  of 
force. — 146.  The  first  problem. — 147.  The  different  forces  ca- 
pable of  producing  a  given  rectilinear  motion. — 148.  The  force 
producing  uniformly  accelerated  motion. — 149.  Law  of  force 
producing  harmonic  motion. — 150.  Law  of  force  producing 
harmonic  motion  \vith  coefficient  of  extinction. — 151.  On  the 
resistance  producing  extinction. — 152.  Law  of  force  producing 
uniform  curvilinear  motion. 

144.  Finite  and  differential  equations    of    motion. — The 

equations  which  define  a  motion  v^^ith  respect  to  three 
coordinate  axes  are  of  the  form 

x=m       y=h(t)       z=h{i) 

where  /,  /i,  and  /2  are  uniform  and  continuous  functions  of  t. 
It  is  only  when  these  three  functions  are  known  that  a 
motion  is  completely  determined. 

The  projections  of  the  velocity  are  given  by 

v.=nt)    Vy=h'{i)    v^^h'it) 

and  the  projections  of  the  acceleration  by 

/x=/"(0  Jy=h"it)  Jz=N'(t) 

123 


124         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Projecting  the  equipoUence 

F  =  mJ 
on  the  three  axes,  we  have  for  the  projections  of  the  force 

F:c  =  mJ:,        Fy  =  inJy        F^^mJ^ 
These  equations,  which  are  usually  written 

ma:"  =  Fx        my"^Fy        mz"  =  F^  [132] 

are  called  the  differential  equations  of  the  motion. 

145.  The  two  general  problems  of  the  mechanics  of  a 
particle  in  terms  of  force. — In  Art.  103  the  general  problem 
of  mechanics  and  its  converse  were  stated  in  terms  of  accel- 
eration, and  it  was  seen  that  if  we  know  the  initial  displace- 
ment and  velocity  of  the  particle  and  the  acceleration  as  a 
junction  of  the  time,  the  motion  is  determined.  The  two 
problems  may  now  be  stated  as  follows: 

1°  Given,  by  observation  or  otherwise,  the  motion — that 
is,  the  equations  of  motion — of  a  'particle,  to  determine  the 
law  of  force. 

2°  Given  the  laws  which  the  forces  acting  obey,  to  deter- 
mine the  motion. 

The  second  problem  is  by  far  the  more  important,  and 
also  the  more  difficult.  However,  the  first  occurs  fre- 
quently enough  to  make  its  study  worth  while.  It  will  be 
seen  that  the  solution  of  the  first  depends  on  the  operations 
of  the  differential  calculus,  which  can  always  be  effected, 
while  the  second  leads  to  the  operations  of  the  integral  cal- 
culus, which  can  be  effected  only  in  a  limited  number  of 
cases. 


DETERMINATION  OF  THE  LAW  OF  FORCE  125 

146.  The  first  problem. — Suppose  the  motion  given  by  its 
finite  equations 

x=m        y=h(t)        z=hit)  [132] 

The  projections  of  the  velocity  are  immediately 

^x=f(0        Vy^h\t)        F.=/2'(0         [133] 

and  of  the  force 

F,=mr(t)        Fy=mh"{t)        F,=mf2'\t)     [134] 

The  force  is  then  determined  in  direction  and  magnitude 
and  the  problem  seems  solved.  In  reality  it  is  not  solved 
if  we  are  looking  for  a  law  of  force  occurring  in  nature, 
for  in  the  components  of  the  force  as  determined  above 
the  time  enters  explicitly,  which  it  never  does  in  a  natural 
law,  inasmuch  as  a  particle  under  the  same  conditions — that 
is,  in  the  same  relative  ^position  with  respect  to  other  par- 
ticles— always  takes  the  same  motion  independent  of  the 
time. 

This  principle  is  termed  the  Principle  of  the  Uniformity 
of  Nature. 

It  is  then  in  terms  of  the  position  and  velocity  of  the 
particle  that  the  force  must  be  expressed.  This  may  be 
done  by  eUminating  t  between  equations  [132]  and  [133] 
giving  one  law  of  force  independent  of  the  velocity,  or 
between  [133]  and  [134]  giving  a  second  law  depending  on 
the  velocity. 

Which,  then,  is  the  natural  law  which  produces  the  motion? 

If  it  is  known  from  physical  considerations  that  the 
force  does  not  depend  on  the  velocity,  as  in  the  case  of  free 


126         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

particles,  then  the  first  law  is  the  one  sought.  Otherwise 
the  problem  is  indeterminate  unless  the  choice  can  be  made 
from  physical  conditions.    Usually  it  can. 

147.  The  different  forces  capable  of  producing  a  given 
rectilinear  motion. — If  the  motion  is  rectilinear,  then  the 
acceleration  and  hence  the  force  is  directed  along  the  path. 

Let  x=f(t) 

be  the  equation  of  motion,  then  the  velocity  is 

and  the  acceleration  is 

The  force  is  therefore 

F  =  mj"{t) 

Eliminatmg  (t)  between  the  first  two  equations,  we  have 
the  law  of  force 

F  =  m<}>{x)  [135] 

This  expression  can  represent  a  law  of  nature,  and  if  we 
know  that  the  force  does  not  depend  on  the  velocity,  it  rep- 
resents the  one  law  capable  of  producing  the  motion. 

But  in  the  absence  of  any  information  to  the  contrary 
we  may  also  eliminate  t  between  the  second  pair  and  get 
the  law  of  force 

F  =  m4>{x')  [136] 

which  can  also  represent  a  natural  law. 


DETERMINATION  OF  THE  LAW  OF  FORCE  127 

Allowing  X  and  x'  both  to  enter,  we  can  get  an  infinite 
number  of  laws  of  force. 

148.  The  force  producing  uniformly  accelerated  motion. — 
Here  there  is  no  ambiguity.  The  acceleration  is  constant 
and  there  is  one  type  of  force  only  capable  of  producing  the 
motion,  namely,  a  constant  force. 

149.  Law  of  force  producing  harmonic  motion. — All  har- 
monic vibrations  of  the  same  period  are  represented  by  the 
equation 

■  x  =  a  sin  {nl  —  ^) 

Whence  2f=an  cos  (nt  —  ^) 

x"  =  —n^[a  sin  (nt  —  0)] 

Hence  there  is  one  solution  only: 

F = mx"  =  -mn^x  [137] 

Therefore,  there  is  hut  one  type  of  force  capable  of  pro- 
ducing all  harmonic  vibrations  of  same  period  in  a  particle.  It 
is  an  attractive  central  force  varying  directly  as  the  distance  of 
the  particle  from  the  centre. 

150.  Law  of  force  producing  harmonic  motion  with  coeffi- 
cient of  extinction. — Consider  the  rectilinear  motion  whose 
equation  is 

X  =  06""' sin  {nt  —  <f>) 

where  e  is  the  base  of  the  Naperian  system  of  logarithms 
and  a  is  a  number,  generally  small  and  essentially  positive, 
which  is  called  the  coefficient  of  extinction. 

For  a  =  0  we  have  the  harmonic  motion,  but  for  a>0 
and  <  =  Qo  we  have  x=0,  which  shows  that  the  motion  dies 
down. 


128         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

1      2A;;r       ,         ,    .  ...  , 

Increasing  t  by  ,  where  A:  is  a  positive  integer,  the 

sine  does  not  change,  but  the  displacement  x  is  multiphed 
by 

2k7:a 

e      " 

and  for  a  sufficiently  large  number  of  periods  becomes  as 
small  as  we  please. 

To  determine  the  law  of  force  capable  of  producing  this 
motion  we  must  calculate  the  acceleration. 

From  the  equation  of  motion 

x'  =  ae~"'[  —  a  sin  (nt  —  (^)  +  n  cos  (nt  —  ^)] 
and 

x"  =ae~'^[  —  {n^  —  a^)  sin  {nt  —  <p) —2an  cos  (nt—<f))] 

Combining  the  first  of  these  with  the  equation  of 
motion, 

ae~"'  sin  (nt  —  4>)  =x 
there  results 

nae  ~"'cos(ni  —  0)=x'  +  ax 

Whence  x"  =  -  (n^  -  a^)x  -  2a  {x'  +  ax) 

and  F = mx"  =  —  m{v?  -\-a'^)x  —  2max' 

151.  On  the  resistance  producing  extinction. —  If  a=0, 
we  have 

F=  —  mn^x 

the  law  of  force  producing  harmonic  motion. 


DETERMINATION  OF  THE  LAW  OF  FORCE  129 

If  a^O,  then  F  may  be  regarded  as  the  algebraic  sum 
of  the  two  components 

which  is  directed  toward  the  centre  and  is  proportional  to 
the  distance  of  the  particle  from  the  centre,  and 

7^2=  -2max' 

of  an  entirely  different  nature.  Instead  of  depending  on 
the  displacement  of  the  particle  it  depends  on  its  velocity 
and  is  always  directed  opposite  to  the  velocity.  It  is,  then, 
always  a  resistance.  It  vanishes  when  the  velocity  is  zero 
and  hence  cannot  produce  motion,  but  has  the  effect  of 
slowing  down  the  motion  produced  by  F\  and  finally  brings 
the  particle  to  rest  at  the  origin. 

152.  Law  of  force  producing  uniform  curvilinear  motion. — 
We  have  seen  that  in  any  curvilinear  motion  the  accelera- 
tion lies  in  the  osculating  plane,  and  its  projections  on  the 
tangent  and  principal  normal  are 


Jt  = 


F2     \dtj 


p 


where  s  is  the  arc  described  measured  from  some  initial 
point  and  p  is  the  radius  of  curvature. 


130         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 
If  the  motion  is  uniform 

K  =  -^  =  K  0,  a  constant 

and  'ft==-T:^=0 

ar 

Hence,   in  uniform   curvilinear   motion   the   acceleration 
is  directed  along  the  normal  and  has  the  value 

P 

The  force  acting  is  then 


F.;„J  =  2^  .         ■       [138] 


directed  along  the  principal  normal,  and  is  therefore  per- 
pendicular to  the  path. 

If  the  path  is  a  circle,  p=R,  the  radius,  and 


CHAPTER  XIII 

MOTION  PRODUCED  BY  FORCES  OBEYING  KNOWN  LAWS 

153.  Statement   of   the   problem. — 154.  Attraction   toward   a  fixed 
centre  proportional  to  the  distance. 

153.  Statement  of  the  problem. — In  the  problem  of  the 
preceding  chapter  we  were  given  the  finite  equations  of 
motion 

x=m        y=h{t)        z=f2it)  [139] 

and  were  required  to  find  the  law  of  force,  independent  of 
the  time,  from  the  projections 

F,  =  mr'{t)        Fy=mh"(t)        F,  =  mh"{t) 

For  this  it  was  sufficient  to  take  the  first  and  second  deriva- 
tives of  /,  /i,  and  J2  with  respect  to  the  time, — a  simple  and 
determinate  problem, — and  then  eliminate  t  between  these 
equations.  We  saw  how  this  could  be  done  in  some  simple 
cases. 

Here,  on  the  contrary,  we  are  given  the  projections  of  the 
force  acting  in  terms  of  the  coordinates,  velocities,  and  the 
time. 

F^{x,y,z,x',i/,zr,t)      Fy{x,y,z,x^,j/,z!,t)      F,(x,y,z,x',y',z\t) 

131 


132         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

and  are  required  to  pass  back  to  the  finite  equations  of 
motion  of  the  particle. 

The  differential  equations  of  the  motion  are  at  once  ob- 
tained by  writing 

mx"  =  F^        my"  =  Fy      '  mz"  =  F^  [140] 

These  equations  [139]  and  [140]  which  define  the  same 
motion  are  equivalent,  but  the  process  of  passing  back  from 
[140]  to  [139]  is  by  no  means  simple,  and  indeed  not  always 
possible. 

As  to  the  force  F  it  may  or  may  not  depend  explicitly  on 
the  velocities  and  the  time.  In  the  case  of  a  free  particle  it 
depends  only  on  the  coordinates — that  is,  on  the  position  of 
the  particle.  In  the  case  of  a  particle  moving  in  a  resisting 
medium  it  will  in  general  depend  on  the  velocities  also,  and 
in  certain  problems  of  constrained  motion  it  depends  explic- 
itly on  the  time. 

Under  these  conditions  the  process  of  passing  from 
[140]  to  [139]  is  one  of  the  most  difficult  operations  of  the 
integral  calculus.  Moreover,  whenever  we  pass  from  a 
derivative  back  to  the  primitive  function  an  arbitrary  con- 
stant is  introduced.  In  this  case,  therefore,  since  there 
are  three  unknown  functions  of  the  time,  namely,  x,  y,  and 
z,  all  entering  to  the  second  derivative,  there  will,  in  gen- 
eral, be  introduced  into  the  solution  six  arbitrary  constants. 
The  problem  is  therefore  very  indeterminate.  The  theory 
of  differential  equations,  however,  tells  us  that,  given  the 
position  and  velocity  of  the  particle  at  any  one  instant, 
there  exists  a  solution,  and  a  single  one.  The  effective  deter- 
mination of  this  solution  is  a  matter  quite  apart  from  its 
existence. 

The  problem  should  then  be  stated  as  follows: 


*N 


MOTION  FROM  FORCES  OBEYING  KNOWN  LAWS      133 

Being  given: 

1°  The  law  of  the  force  or  forces  acting  on  the  'particle, 

2°  The  initial  -position  and  velocity  of  the  particle, 
determine  its  motion,  that  is,  its  position  at  each  instant. 

154.  Attraction  toward  a  fixed  centre  proportional  to 
the  distance. — A  special  problem  will  illustrate  the  difficul- 
ties and  also  the  method  of  procedure. 

Let  (fig.  35)  0  be  a  fixed  point  and  M  a  particle  of  mass 
m  acted  on  by  a  force  F  always  directed  toward  0  and  pro- 
portional to  MO. 


Fig.  35. 

Let  Mo  be  the  initial  position  of  M  and  Vo  its  initial  veloc- 
ity. Take  the  X-axis  through  Mq  and  the  F-axis  in  the 
plane  of  OX  and  Vq,  the  Z-axis  being  then  perpendicular  to 
this  plane. 

Let  X,  y,  z  be  the  coordinates  of  M  at  the  instant  t, 

then  mx"==F^        my"  ==Fy        mz"=F, 

Denoting  by  n^  a  constant,  we  may  write 

F  =  m-n^-r 


134         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 
where  r  denotes  the  distance  OM. 


Then       F^=~F- 
r 


Fy=-F-^ 


and  hence 


Fx=  —m-n^-x 
Fy=  —m-rfi-y    ■ 
Fz=  —m-n^-z 


F,=  -F- 
r 


The  differential  equations  of  motion  are  then 


r"  =  'm2-i 


X  =  —n^x 


>,"=-n^y 


"=  _^2 


nfz 


[141] 


The  first  of  these  contains  x  only,  the  second  y  only,  and  the 
third  z  only,  and  may  therefore  be  studied  separately.  This 
is,  in  general,  not  the  case.  If,  for  example,  F,  instead  of 
varying  as  r,  varied  inversely  as  r^,  we  should  have 


F== 


rn-v/' 


and 


F. 


m-n^-x 


In  this  case  each  differential  equation  of  motion,  instead 
of  containing  one  coordinate  only,  would  contain  the  dis- 
tance 


r  =  \/x^+y^+z^ 

to  the  —  ^/2  power.    These  equations   could   not   then   be 
treated  separately,  but  would  form  a  system  of  simultane- 


MOTION  FROM  FORCES  OBEYING  KNOWN  LAWS      135 

ous  differential  equations  in  three  unknowns,  and  the  diffi- 
culty of  finding  a  solution  would  be  immensely  increased. 
To  return  to  equations  [141],  the  equation 

x"=-nH  [142] 

expresses  that  in  the  projection  of  the  motion  on  the  X-axis 
the  acceleration  is  always  directed  toward  0  and  is  propor- 
tional to  the  abscissa  of  the  projection  of  M  on  the  X-axis. 
But  we  have  seen  that  every  harmonic  motion  along  OX 

with  0  as  centre  and  of  period  —  has  exactly  this  equation, 

no  matter  what  the  amplitude  a  and  the  phase  ^  may  be. 
Hence,  the  finite  equation  of  motion 

X  =  a  sin  {nt  —  </>) 
is  a  solution  of 

x"  =  —n^x 

no  matter  what  values  the  constants  a  and  <f>  may  have. 
Putting  —asm(f)^A,  acos(f)=B,  we  may  write 

x  =  A  cos  nt+B  sin  nt'  '  [143] 

In  fact  we  have 

x'  =  /i[ — A  sin  nt  +  B  cos  nt] 

x"  =  -  n^[A  cos  nt  +  B  sin  nt]  =  —  nH 

which  shows  that  [143]  satisfies  [142].     In  the  same  way 

y=:  A\  cos  nt+Bx^xnnt  [144] 

z=A2  cos  nt+BiSm.  nt  [145] 


136 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


where  Ai,  B\,  A2,  B2  are  arbitrary  constants.  These  are 
then  the  finite  equations  of  motion,  containing  six  arbi- 
trary constants,  which  must  be  so  determined  as  to  satisfy 
the  initial  conditions 


x  =  OMo  =  Xo         y  =  0  0  =  0 

^  =  ^ox  y'  =  Voy        z'  =  Yqz 

for  ^  =  0. 

Since  z  and  z'  vanish  for  f  =  0,  it  is  easily  shown  that 
z  is  identically  zero,  and  the  motion  is  in  the  plane  oiXY . 

In  fact,  from  [145] 


z'  ='Y^—A2  sin  wf  4-52  cos  ni\ 


whence,  for  <  =  0 

2  =  ^2  =  0  0'=  -71^2  =  0 

and  therefore  ^2  =  ^2  =  0. 

We  have,  then,  only  to  consider  the  equations  in  x  and  y. 
For  i  =  0  we  have 


x  =  A  =a:o        x' =nB  =Fox 
y  =  Ax^  0        y'  =  nBx  =  Y^ 
hence  the  equations  of  motion  are 

Fo 


X'=xn  COS  ni +— ^  sm  ni 

n 

V= smnt 


[146] 


Two  cases  present  themselves  according  as  the  initial 
velocity  T^o  has  or  has  not  the  direction  of  OX. 


MOTION  FROM  FORCES  OBEYING  KNOWN  LAWS      137 

In  the  first  case 

and  the  equations  of  motion  are 

x  =  Xq  cos  nt^ sin  nt 

n 

the  A'elocity  Vo  being  positive  or  negative  according  as  it 
is  directed  away  from  or  toward  0. 

The  motion  is  rectilinear  and  harmonic,  and  the  ampH- 
tude  and  phase  are  easily  deduced. 

In  the  second  case  let  Vq  make  an  angle  a  with  the  posi- 
tive direction  of  the  X  axis, 


then 
whence 


Vqj:  =  Vo  cos  a         Voy  =  Vo  sin  a 


B  = 


Vo  cos  a 


Bi  = 


Vq  sin  a 


and 


Vo  cos  a   . 

x=Xo  cos  nt-\ smnt 

n 

V'osina   . 
V  = smnf 


The  equation  of  the  path  obtained  by  eliminating    t 
between  these  two  equations  is 


2-*)2 


Xo'-n 


(x  sin  «  - 1/  cos  a)  2  +  „     y^  =  xo^  sin^a 

r  0 

which  is  easily  seen  to  be  an  ellipse  whose  diameter  con- 
jugate to  OX  is  parallel  to  Vo. 


138         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

If  a  =  90°,  and  hence  Vo  perpendicular  to  OX,  the  path  is 
the  elUpse 


Observe  that  from  equations  [146] 


Fo 


whence 


x'  =  n{  —Xo  sin  nt-\ cos  nt 


')! 


y'  =  Vqv  cos  nt 


Similarly 


— 5 + x2  =  — 5^  +  xo^  =  constant 
n2  n^ 


/2 


+,-%^constant 


Adding,  and  observing  that 


we  have 


x^  +  y^  =|.2 


72  +  ^2^.2  =  7p2  4.  ^22;o2  =  constant 


which  shows  that  the  velocity  V  of  the  particle  M  depends 
only  on  its  distance  r  from  0,  the  centre  of  attraction. 

This  is  a  particular  case  of  a  more   general   theorem, 
known  as  the  theorem  of  Vis  Viva. 


MOTIOiX  FROM  FORCES  OBEYING  KNOWN  LAWS      139 
From  the  first  two  of  equations  [141] 


we  see  that 

Hence 

and  therefore 


yx"-xy"=0 
Areal  /  =  0 

Areal  V  =  constant 


The  elhptic  path  is  then  described  under  the  law  of  constant 
areal  velocity. 

To  resume : 

1°  In  the  most  general  case,  that  is,  whatever  the  initial 
conditions,  an  attraction  toward  a  fixed  centre  propor- 
tional to  the  distance  produces  a  periodic  elliptic  motion, 

2?: 
whose  period  is  7"  =—,  where  n^  is  the  coefficient  of  attrac- 
tion. 

The  ellipse  has  its  centre  at  the  centre  of  attraction,  and 
admits  the  directions  of  the  initial  radius  vector  and  veloc- 
ity as  conjugate  directions;    the  semiconjugate  diameters 

Vo     VqT 
are  equal  to  the  initial  radius  vector  xq  and  the  ratio  —  =  -x — • 

of  the  initial  velocity  Vq  of  the  particle  to  the  angular  veloc- 

ity  w  =  ^  corresponding  to  the  period  of  the  elliptic  motion. 

2°  In  the  case  where  the  initial  velocity  is  perpendicular 
to  the  initial  radius  vector,  these  two  conjugate  diameters 
become  the  axes  of  the  ellipse. 


140        ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

3°  If,  moreover,  the  relation 

Vo=nxo 

holds  between  the  initial  velocity  Vo  and  the  initial  radius 
vector  Xo  and  the  angular  velocity  n — that  is,  if  the  initial 
velocity  Vo  corresponds  to  the  angular  velocity  n — the 
motion  is  circular  and  uniform  with  angular  velocity  n. 

4°  If  the  initial  velocity  coincides  in  direction  with 
the  initial  radias  vector,  the  motion  reduces  to  harmonic 
rectilinear  motion. 

5°  In  every  case  the  motion,  either  elliptic  or  circular, 
may  be  considered  as  the  resultant  of  two  harmonic  motions 
of  the  same  period  along  two  directions,  oblique  or  rectan- 
gular. 

6°  The  velocity  has  the  same  magnitude  each  time  the 
particle  is  the  same  distance  from  the  origin. 

7°  The  path  is  always  described  under  the  law  of  con- 
stant areal  velocity. 

Remark. — ^This  attractive  force  proportional  to  the  dis- 
tance arises  whenever  a  system  is  slightly  displaced  from 
its  position  of  equilibrium.  We  shall  find  an  important 
application  in  the  pendulum.  Other  applications  are  fre- 
quent in  optics,  acoustics,  and  electricity. 


CHAPTER  XIV 

RELATIVE  MOTION      . 

155.  Relative    axes. — 156.  Relative    force. — 157.  Problems   of   rela- 
tive motion. — 158.  First  problem. — 159.  Second  problem. 

155.  Relative  axes. — In  the  previous  chapters  we  have 
referred  the  motion  to  the  absolutely  fixed  axes.  In  certain 
classes  of  problems  this  is  inconvenient.  For  instance,  in 
considering  the  motion  of  the  planets  about  the  Sun,  it  is 
convenient  to  refer  these  motions  to  a  set  of  axes  having 
the  origin  at  the  centre  of  the  Sun  and  remaining  parallel 
to  themselves.  For  the  motion  of  particles  on  or  near 
the  Earth's  surface  it  is  convenient  to  refer  to  a  set  of  axes 
fixed  on  the  Earth,  and  in  this  case  the  origin  is  not  only 
not  fixed,  but  the  directions  of  the  axes  change  as  the  Earth 
moves  in  its  orbit  and  about  its  axis.  These  are  the  rela- 
tive axes. 

156.  Relative  force. — I  shall  define  the  relative  force 
as  the  product  of  the  mass  by  the  relative  acceleration,  that 
is,  by  the  equipollence 

Tr^m-i;  [147] 

157.  Problems  of  relative  motion. — The  problems  of  rela- 
tive motion  ma}-  be  stated  in  a  manner  analogous  to  those 

of  absolute  motion. 

141 


142         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

1°  Given  the  relative  motion  of  a  particle  to  determine 
the  law  of  force. 

2°  Given  the  law  of  force  and  the  initial  conditions  to 
determine  the  relative  motion. 

These  problems  would  play  the  same  role  and  could  be 
attacked  in  the  same  way  as  the  corresponding  problems  in 
absolute  motion  if  we  had  a  set  of  principles  for  relative 
motion.  It  is  a  priori  evident,  however,  that  the  absolute 
motion  being  determined  by  these  principles,  the  relative 
is  also,  and  it  becomes  simply  a  question  of  restating  them 
for  the  relative  axes. 

158.  First  problem. — Such  a  restatement,  while  not  very 
difficult,  is  neither  necessary  nor  of  advantage,  for,  in  the 
case  of  the  first  problem,  if  the  equations  of  motion  be 

$  =  cl>{t)  r;  =  <^i(0  :  =  <j>2(t) 

the  relative  velocity  and  acceleration  can  be  obtained  by 
differentiation,  and  one  or  more  laws  of  relative  force  are 
arrived  at  by  eliminating  t,  and  this  entirely  independent 
of  the  motion  of  the  relative  axes. 

If  it  is  desired  to  obtain  the  law  of  absolute  force  from 
the  equations  of  relative  motion,  the  motion  of  the  rela- 
tive axes  must  also  be  given.  Suppose  this  done,  that  is, 
suppose  0,  Oi,  02;  a,  b,  c;  ai,  bi,  ci;  a2,  62,  C2  to  be  given 
functions  of  the  time,  the  absolute  force  is  then 

mJ^=m(J^+J^+7t)  [148] 

where  the  projections  of  Jc,  Jr  and  Jt  are  given  in  Art.  100. 
Eliminating  t  between  this  equipollence,  the  equipollence 


RELATIVE  MOTION  143 

and  the  equations  of  motion 

e  =  </>(<)       v  =  Mt)       C=^2(0 

we  have  the  various  possible  laws  of  absolute  force  ex- 
pressed either  in  terms  of  the  absolute  or  the  relative  coor- 
dinates and  velocities,  as  we  please. 

159.  Second  problem. — Given  the  law  of  relative  force 
and  the  initial  relative  conditions,  the  second  problem  is 
reduced  to  the  integration  of  the  differential  equations  of 
relative  motion: 

m^"  =  Fr^        mi'  =  Fr^        mC"  =  Fr^         [149] 

If,  on  the  other  hand,  it  is  the  law  of  absolute  force  which 
is  given,  and  the  relative  motion  is  desired,  then  the  rela- 
tions of  Art.  100  must  be  made  use  of. 

From 

we  have  at  once 

¥r  =  mJ^=m{J^-  Tc-T^=m  {-^  -T^  -TS     [150] 

where  J  a,  Jc,  and  Jt  are  here  to  be  expressed  in  terms  of  the 
relative  coordinates  and  velocities  by  means  of  the  relations 

x  =  o+a$  +  bT^  +  c{^ 

y  =  Oi+ai$+birj  +  ci!^  ■ 

t/'  =  (oi'+ai'$  +  6i'i?  +  ci'c)  +  (air  +  6iV+ciC') 
2'  =  (02  +  a2'^  +  62'5?  +  C2'C)  +  (02^  +M'  +C2C') 


144         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

The  differential  equations  of  relative  motion  then  become 


m^"  ^Fat—fnJct—mJt^ 
m-q"  ==Fa^-  mJc^  -  mJt^ 
in(^"  =  Fa.  —mJcr  —''nJtr 


[151] 


and  the  problem  is  reduced  to  the  integration  of  these  equa- 
tions. The  right-hand  members  will  in  general  contain 
the  time  explicitly,  so  that  the  equations  [151]  are  of  the 
form 

r=^ii^,V,C,^',v',C',t)   ■  [152] 


CHAPTER  XV 

MOTION  OF  A  FREE  PARTICLE  ON  EARTH'S  SURFACE 

160.  The   relative   axes. — 161,  Differential   equations   of   motion. — 
162.  First  approximation. — 163.  Second  approximation. 

i6o.  The  relative  axes. — Refer  the  motion  to  a  set  of 
axes  having  the  origin  at  a  point  A  near  the  Earth's  surface 
and  orientated  in  the  following  manner:  Take  the  axis  of  (^ 
along  the  direction  of  the  geometric  difference  of  the  absolute 
and  convective  acceleration  of  A,  the  axis  of  ^  in  the  plane 
of  the  meridian  through  A  drawn  toward  the  south,  and 
the  axis  of  t]  toward  the  east.  Denote  by  g  the  geometric 
difference  of  the  absolute  and  convective  accelerations  of  any 
point  M{^,  7j,  C).  The  acceleration  g  will  then  depend, 
both  in  magnitude  and  direction,  on  the  relative  coordi- 
nates $,  r),  and  (^  of  M,  but  for  a  first  approximation  may  be 
taken  as  constant  and  equal  to  its  value  at  A,  since  for  any 
region  about  A  in  which  the  motion  of  a  particle  on  the 
Earth's  surface  is  considered  its  variation  is  of  the  order 
of  (t?,  where  to  is  the  angular  velocity  of  the  Earth. 

Now,  expressed  in  radians  per  sidereal  second 

w  =  24^  =  . 0  00073+  to2  =  .000  000  0053  + 

and  hence  terms  of  the  order  of  a?  may  be  neglected  to  a 
first  approximation.    The  tlirection  of  t^  is,  hy  definition, 

145 


146 


ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


the  vertical  at  A.  Suppose  the  moving  particle  at  any 
instant  to  have  the  relative  coordinates  ^,  tj,  r.  The  pro- 
jections of  the  turning  acceleration  on  the  relative  axes  are 


•^T 


26r^Sin  X 
at 


dc 
dt 


Jt  =2a>[-T7  sin  ^—^7  cos  X 

T       ^   din         . 
J,  =2iij-Y^  cos  A 
'c         dt 


[153] 


where  X  is  the  latitude  of  ^. 

i6i.  Differential  equations  of  motion. — Substituting  these 
values  in  the  differential  equations  of  motion  of  Art.  159 
we  have 


^   drj  .     , 

■■2co-Tr  sm  / 
dt 

/de  . 


dc 


C"=^-2^cos^ 


[154] 


The  integration  of  these  equations  gives  the  finite  equa- 
tions of  relative  motion  in  terms  of  six  arbitrary  constants, 
which  are  determined  by  the  initial  position  and  velocity. 

162.  First  approximation. — Neglecting  the  terms  con- 
taining oj  for  a  first  approximation,  the  differential  equa- 
tions of  motion  become 

that  is,  the  particle  is  acted  on  by  an  approximately  con- 


MOTION  OF  FREE  PARTICLE  ON  EARTH'S  SURFACE  147 

stant  relative  force 

F2=mg 

directed  along  the  vertical  toward  the  interior  of  the  Earth. 

This  force  is  termed  the  weight. 

163.  Second  approximation. — Retaining  the  terms  con- 
taining CO  and  assuming  the  particle  to  start  from  the  origin 
with  zero  relative  velocity,  we  have  by  integrating  equa- 
tions [154]  once 

^'  =  2(1)7}  sin  X  "I 

rf  =  -2w(e  sin  >i  -  c  cos  X)  y  [156] 

(^' =gt -2oj7j  cos  X  J 

Attempting  to  satisfy  these  equations  by  functions  of 
the  form 


^  =  ^0  +  ^^1+^2^2+1 
7j  =  1^0  +  OJTji+O?7)2  + 


[157] 


where  ^0,  Vo,  Co>  ^i>  Vi}  C>  etc.,  are  functions  of  t  which  van- 
ish together  with  their  first  derivatives  for  t  =  o,  we  have  by 
substituting  these  developments  in  equations  [156]  and 
equating  like  powers  of  a» 


whence 


Similarly 


d^o  dr)o  d(^o     _^ 

-du^'     dr=^    -w^^ 


d^i  dtji  d^i 


148         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

whence 

ci=o        j;i=— cos/        Ci=o 

Neglecting  the  terms  in  oJ^  in  the  development  [157]  we  have 

qt^  qt^ 

e=o        ri  =  oj-jcos}.        c  =  y         [158] 

to  the  second  approximation. 

The  particle  then  remains  in  the  plane  of  j?C,  but  instead 
of  falling  along  the  vertical  it  is  deviated  slightly  toward 
the  east.  In  the  r^C  plane  the  particle  describes  the  semi- 
cubical  parabola 

3\/2g-r)  =  4:Oj^'^^cos  X 

which  gives  the  eastern  deviation   for  a  fall  from  a  given 
height. 

Retaining  terms  in  cu^  in  [157]  we  should  get  a  southern 
deviation,  but  since  the  variation  in  g  is  also  of  the  order 
of  if2  this  method  will  not  give  its  numerical  value. 


CHAPTER  XVI 

CONSTRAINED  MOTION  OF  A  PARTICLE.     GENERAL 
CONSIDERATIONS 

164.  Constrained  particle. — 165.  Mathematical  expression  of  the  con- 
ditions of  constraint. — 166.  Variable  constraints. — 167.  Smooth 
constraints. — 168.  Forces  of  constraint. — 169.  Motion  of  a  con- 
strained particle. — 170.  Direction  of  Fr. — 172.  Magnitude  of 
Fr. — 173.  Particle  at  rest  on  a  smooth  surface. — 174.  Curve  of 
constraint. — 175.  Particle  at  rest  on  a  smooth  curve. — 176.  Con- 
■  ditions  of  equilibrium. 

164.  Constrained  particle. — A  particle  is  said  to  be  con- 
strained if  it  is  prevented  by  the  physical  conditions  of  the 
problem  from  occupying  certain  positions  in  space,  or  if  it 
is  prevented  from  passing  from  one  position  to  another 
along  certain  paths.  For  example,  a  particle  attached  to 
a  fixed  point  A  by  an  inextensible  string  of  length  I  can 
never  reach  that  portion  of  space  outside  of  the  sphere 
of  radius  I  about  A  as  centre.  Similarly  a  particle  may 
be  constrained  to  lie  on  a  surface  {S)  or  on  a  curve  (C). 

165.  Mathematical  expression  of  the  conditions  of  con- 
straint,— In  every  case  the  conditions  of  restraint  may  be 
expressed  by  mathematical  relations  among  the  coordinates 
of  the  particle.  In  the  first  example  above  the  condition 
may  be  expressed  by  writing 

14'J 


150         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

where  oi,  02,  as  are  the  coordinates  of  A,  and  in  general, 
if  the  particle  is  required  to  lie  within  or  without  a  sur- 
face (S)  we  may  write  the  condition  that  the  distance  of 
the  particle  from  some  point  within  (-S)  shall  be  less  or 
greater  than  the  line  from  this  point  to  the  surface. 

If  the  particle  is  required  to  lie  on  the  surface  (S),  then 
at  every  instant  its  coordinates  must  satisfy  the  equation 
of  the  surface 

S  =  0 

If  it  is  required  to  lie  on  the  curve  (C),  then  its  coor- 
dinates must  satisfy  the  equations  of  the  curve. 

The  physical  conditions  are  thus  replaced  by  mathe- 
matical equations. 

166.  Variable  constraints. — If  the  constraining  surfaces 
or  curves  change  their  form  or  position  or  both  with  the 
time,  they  are  termed  variable,  and  their  equations  will 
involve  the  time.  These  are  excluded "  from  the  following 
discussion. 

167.  Smooth  constraints. — I  shall  assume  the  constraints 
to  leave  unaltered  the  component  of  the  acceleration  tan- 
gential to  them.  They  are  then  termed  smooth,  or  without 
friction. 

168.  Forces  of  constraint. — In  addition  to  the  con- 
straints the  particle  may  be  acted  on  by  certain  forces, 
termed  the  external  forces.  Let  Fg  be  the  geometric  sum 
of  these  external  forces  and  let  /«  be  the  corresponding 
acceleration,  so  that 

Fl=mT,  [159] 

The  particle  under  the  action  of  Fg  and  the  constraints 
will  take  a  certain  motion,  and  will  therefore  at  each  instant 


CONSTRAINED  MOTION  OF  A  PARTICLE  151 

have  a  certain  velocity  and  acceleration.  Let  V  and  J 
represent  these  and  wi-ite 

J-J^^Tr  [160] 

I  shall  term  Jy  the  acceleration  of  the  constraints  and  F^i, 
where 

the  force  of  the  constraints. 

169.  Motion  of  a  constrained  particle. — It  follows  that 
the  motion  of  the  constrained  particle  will  be  the  same  as 
that  of  a  free  particle  acted  on  by  a  force  F,  where 

F  =  Te+F~R 

Hence,  given  Fg  and  the  constraints  the  problem  is  reduced 
to  the  determination  of  Fr,  or  conversely,  given  the  motion 
and  the  constraints,  to  the  determination  of  F  and  thence 
Fe  and  Fr,  in  a  form  not  involving  the  time  explicitly. 

170.  Direction  of  Fr.  Surface  of  constraint. — The  tan- 
gential components  of  Jg  are  unaltered.     Hence 


Ji=Je, 

But 

Jt=Jet+JRt 

Therefore 

^«,=o 

[161] 

and,  the  acceleration  of  constraint  is  normal  to  the  surface  of 
constraint. 


152         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

171.  Magnitude  of  Fr, — Projecting  on  the  normal 

Hence  Jr=Jn-  Je^  [162] 

and  Fr  =  Fx-F;^  [163] 

172.  Particle  at  rest  on  a  smooth  surface. — If  the  particle 
is  at  rest 

J=0 

and  Jr=  -Jejf 

Hence,  the  reaction  of  a  smooth  surface  on  a  particle  at 
rest  on  it  is  equal  and  opposite  to  the  projection  of  the  resultant 
of  the  external  forces  on  the  normal. 

173.  Curve  of  constraint.  —  As  before,  the  tangential 
component  of  Jg  is  unaltered.     Hence 

Whence  "^Rt^^ 

and  therefore,  the  reaction  is  in  the  normal  plane  to  the  curve. 
Projecting  on  the  normal  and  the  binormal 


Jn—Jcj^+Jr^ 


JB  —  Je^  +  jRg  =  0 


CONSTRAINED  MOTION  OF  A  PARTICLE  153 


whence  J^  ^  =  Jn  —  J, 


ejv 


JrS Je 


[164] 


which  give  the  components  along  the  normal  and  binormal 
and  hence  determine  Jr. 

174.  Particle  at  rest  on  a  smooth  curve. — ^If  the  particle 
is  at  rest,  then 

J-0=Je    1 

and  JRf^=-JeN  [  [165] 

JrB^  ~'^es    J 

Therefore,  for  a  particle  at  rest  cm  a  smooth  curve  the 
reaction  is  equal  and  opposite  to  the  external  force. 

175.  Conditions  of  equilibrium. — Hence,  in  order  that 
a  particle  be  in  equilibrium  on  a  smooth  surface  the  external 
force  must  be  diverted  along  the  normal,  and  in  order  that 
it  be  in  equilibrium  on  a  smooth  curve  the  external  force 
must  lie  in  the  normal  plane. 


CHAPTER  XVII 

CONSTRAINED  MOTION  ON  EARTH'S  SURFACE 

176.  Variation  of  g  with  latitude. — 177.  Particle  falling  on  smooth 
curve. — 178.  Oscillations  of  a  simple  pendulum. — 179.  Oscil- 
lations of  small  amplitude. — 180.  Velocity  in  oscillations  of  any 
amplitude. — 181.  Tension  in  the  string. — 182.  Foucault's  pen- 
dulum.— 183.  Approximate  solution.  Particular  case.  General 
case.    Foucault's  case.    Theorem  of  Chevilliet. 

176.  Variation  of  y  with  latitude. — In  Art.  160  the  direc- 
tion of  the  vertical  at  a  point  A  was  defined  as  the  direction 
of  the  geometrical  difference  of  the  absolute  and  convec- 
tive  acceleration  of  A,  and  the  magnitude  of  this  difference 
was  denoted  by  g,  termed  the  acceleration  of  gravity  at  A. 
This  acceleration  is  then  defined  by  the  equipollence 

'g=Ta-Ta  [166] 

Neglecting,  for  a  first  approximation,  the  attraction  of  the 
sun  and  the  moon  and  the  orbital  motion  of  the  Earth 
about  the  Sun,  and  assuming  the  Earth  to  be  a  sphere  com- 
posed of  concentric  homogeneous  layers,  J  a  is  at  any  instant 
directed  along  the  line  joining  A  with  the  centre  0  of  the 
Earth,  Denote  this  value  of  /„  by  go,  and  choose  the  abso- 
lute axes  as  follows:  X  and  Y  in  the  plane  of  the  equator, 
and  Z  through  0,  coincident  with  the  axis  of  rotation  of 
the   Earth,   positive    direction .  toward   the   north.    Orient 

154 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE 


155 


the  relative  axes  as  in  Art.  160,  that  is,  C  along  the  direc- 
tion of  g,  H  in  the  meridian  toward  the  south,  and  ij  toward 
the  east.  Denote  by  co  the  angular  velocity  of  the  Earth 
and  by  a  its  radius,  and  let  r  (fig.  36)  be  the  radius  of  the 


Fig.  36. 


parallel  on  which  A   lies.    Then  Jc  is  constantly  directed 
along  r  and  its  magnitude  is 


Jc  =  ror  =  aoj^  cos  Xq 


[167] 


where  -^o  is  the  geocentric  latitude  of  A.  Denoting,  as 
before,  by  X  the  astronomical  latitude  of  A,  and  projecting 
the  equipollence 


g=go—aoj^  cos  ^o 
on  A(^,  we  have 

g  =  go  cos  {X  —  Xq)  —aco^  cos  Xq  cos  X 


[168] 


The  angle  X  —  ^o,  termed  the  equation  of  the  vertical,  may  to 
a  first  approximation  be  taken  as  zero, 


i.:g      elements  of  kinematics  and  mechanics 
whence  g  =  go  —  aoj^  cos^  k  [169] 

From  observations 

g  =  980.6059  cm.  -  2.5028  cm.  cos2  X  [170] 

177.  Particle  falling  on  a  smooth  curve. — If  a  particle 
M  is  constrained  to  describe  a  curve  (C)  under  the  action  of 
an  external  force  Fg,  obeying  a  known  law,  the  problem  is 
to  determine  the  equation 

giving  the  arc,  measured  from  some  initial  point,  in  terms 
of  the  time. 

The  equation  of  motion  may  also  be  determined  in  the 
form 

where  it  is  solved  for  t 

As  we  have  seen,  the  reaction  N  of  the  constraint  is 
always  in  the  normal  plane  to  the  curve,  and  M  may  be 
regarded  as  free  under  the  action  of  the  two  forces  N  and 

Since  the  tangential  acceleration  is  unaltered 


inJt  =  F, 


cPs 
"'dP  =  ^',  [171] 


which  determines  s  in  terms  of  t  when  the  initial  conditions 
are  known. 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE 


157 


If  the  external  force  is  the  force  of  gravity  and  the  rela- 
tive axes  be  orientated  with  Z  drawn  along  the  vertical 
]rom  the  interior  of  the  Earth  (fig.  37),  and  if  7  be  the 


velocity  of  M  at  any  point,  and  0  the  angle  between  the 
tangent  at  M  and  the  axis  AZ,  then 

r' 

cos    0=y 


whence 


Fet=-mgy 


or 


mV  =  —mQy 


Hence 


or 


VV'=-gC 
VV'+g:'=0 


which  m"ay  be  written 


158         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Integrating 

-w  +gC='C    a  constant 

If  then  (^0  is  the  initial  altitude  and  Vq  the  initial  velocitjf^ 

or  ^2  =  Fo2  +  2^(Co-C)  [172] 

which  gives  the  velocity  of  M  when  it  passes  any  point  of 
its  path  of  altitude  (^. 

It  is  to  be  noted  that  this  velocity  depends  only  on  Vq 
and  Co  — C>  that  is,  on  the  initial  velocity  and  the  difference 
in  altitude  of  the  given  point  and  the  initial  point,  and 
not  at  all  on  the  form  of  the  curve  (C). 

Since  ((7)  is  supposed  given,  then  C  is  a  known  function 
of  s,  and  hence 

V  =  ±  VFo2  +  2gr(Co-0  =  <;6i(s) 

whence  j^  =  ^i(s) 

ds 
or  dt 


Ms) 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        159 


ds 


Integrating  t  =  J^^  ^-^  =  ^is)+C 


=  <jE.(s)-</>(so) 


[173] 


and  the  problem  is  solved. 

178.  Oscillations  of  a  simple  pendulum. — A  simple  pen- 
dulum is  a  particle  M  constrained  to  describe  a  circumfer- 
ence of  radius  l=AM  under  the  action  of  gravity  (fig.  38). 


A 

/V 

"\ 

i> 

y 

k 

MN^ 

j^ 

z 

Fig.  38. 


Suppose  the  relative  axes  orientated  with  AZ  along  the 
vertical  toward  the  interior  of  the  Earth,  and  AS  in  the 
plane  of  AZ  and  Mo,  the  initial  position  of  the  pendulum. 
//  we  neglect  the  rotation  of  the  Earth,  there  is  the  constant 
force 

F,  =  mg 


acting  parallel  to  AZ,  and  hence  the  pendulum  remains  in 
the  plane  3AZ. 


160         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

Let  d  be  the  angle  which  AM  makes  with  AZ  at  any 
instant  t,  and  let  the  pendulum  be  at  its  highest  position 
at  the  instant  t=Q.  Count  s  from  the  lowest  position,  posi- 
tive when  ^  is  positive. 

Then  s=ld  V  =16'  V  =W" 

!:=lcosO     :'  =  -lsmd-0' 

whence  P-d'-  0"  =  -  gl  sin  0  ■  0' 

or  .     r=— |sin^  [174] 

This  is  the  differential  equation  of  the  motion. 
Integrating  once 

Now  when  6=0^,  then  6'  =-0q  ,  and  hence 


c  = 

20"''- 

-  y  COS  Oq 

=  J«o-  +  ^( 

cos  6- 
dd 

-cos 

^o) 

sl»o' 

'^.f( 

COS  6  - 

-COS 

^o) 

Therefore  6'  =  ^J^o'2 + ^(cos  6  -  cos  ^o)  [175] 

whence 


or  t=   I     ,  ^^  +c     [176] 

|V2+-^(cos^-cos^o) 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        161 

179.  Oscillations  of  small  amplitude. — The  integration  of 
this  expression  leads  to  elliptic  functions,  but  a  simplifica- 
tion may  be  introduced  if  we  assume  ^0  so  small  that,  for 
a  first  approximation,  sin  6  may  be  replaced  by  6  in  [174]. 
In  this  case  the  differential  equation  of  motion  becomes 


and  the  motion  is  harmonic. 
Integrating 


d  =  A  cos  \j-t  +  B  sin  ^j^^t  [177] 

whence      V  =  16'  =  1^^^- A  sin  ^^-t  +  B  cos  -yj^-t] 


If  the  pendulum  is  abandoned  at  rest,  then  forY=0  we 
must  have  7  =  0,  whence  J5=0,  and 


d=^A  cos  J-4--t 


Since  for  t=0,  6  =  60,  we  have 


O  =  0ocos^^'t 
and  e'=-^o^j'^■sm^^'^ 


[178] 


162         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

The  period  T  is  such  that  replacing  thy  t-\-T  both  0  and 
d'  take  their  original  values, 


whence  ^y-T  =  27c 

-4 


or  7' =  2      '^ 


i8o.  Velocity  in   oscillations   of   any  amplitude. — Substi- 
tuting C  =  ^  cos  6  in  [172]  we  have 

72  =  y^,2_^2(/i(cos^-cos^o)  [179] 

which  agrees  with  [175],  and  gives  the  velocity  at  any  point 
of  the  path  for  any  amplitude. 

i8i.  Tension  in  the  string. — From  the  equations  of  Art. 
168 


we  have  at  once 


Jrb-  ~JeB 


V2 

jR^=-l^+g  COS  d 


Jrb-^ 


Hence  the  tension  in  the  string  is 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        163 
T=in-j-  +  mg  cosd 


=  m—j—  +  mg(3  cos  6  —  cos  ^o)  [180] 


The  maximum  value  of  T  is  when  the  pendulum  passes 
through  the  lowest  point  of  its  path,  or 


T'maximum  =  — ^  +  W^  (3  -  COS  do)  [181] 


182.  Foucault's  pendulum. — If  the  rotation  of  the  Earth 
be  taken  into  account,  then  the  external  relative  force  is 


Fre^  =  —  niJt^ 
Frr^=-mJt^ 
Fre^  =  mg-mJt^ 

The  particle  is  constrained  to  lie  on  the  sphere  . 

c2  +  ,2  +  ^2=;2 

and  calling  N  the  reaction  of  this  smooth  surface,  we  have 


N,=  -N' 


N^^-N.f- 


Nc—N-f 


since  N  is  directed  toward  A. 

The  relative  force  acting  is  then 


164         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


Fr^-^-mJ,^-Nj 


Fr  =-mJ,  -N-i 


Fr^=mg-mJt^-Nj 


and  the  differential  equations  of  relative  motion  are 


m7)"=-mJt^-N-f 

r 
mc"  =  mg  -  mJt,  -N-j- 


Replacing  the  components  of  the  turning  acceleration 
by  the  expressions  of  Art.  161  we  have  finally 


m^"=-2m(o[  -37  sin  X+-:^  cos  XJ -N-i- 


dt 


dt 


mi{'  =  —  2ma>-^  sm  X—N-j- 
m!^"=mg  +  2mct/-j-  cos  X—N-j- 


[182] 


These  three  simultaneous  differential  equations  together 
with  the  equation  of  constraint 


e2  +  ry2  +  C2=;2 


[183] 


give  the  motion. 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        165 


In  order  to  integrate  [182]  the  reaction  of  constraint  N 
must  be  eliminated  by  means  of  [183].  This  may  be  done 
as  follows: 

Solving  [183]  for  (^  we  have 


Differentiating 


^     =V;2_^2_^2 


c'  ={\/p-$^-fy 


[184] 


Substituting  in  the  third  equation  of  [110]  we  have 


m {\/P  —  f 2  _  ,j2) "  =yYig^ 2mcoT/  cos  X—N 


I 


"\Mieiice 


N- 


ml 


;j^^^==^\g-2coi  ZOS  X-Wl^-^^-yf)"\ 


Substituting  these  values  of   ^  and  N  in  the  first  two 
equations  of  [182]  we  have 


Tj"  =  -2w)  e'  sin  yl -  Wl^-^^-ri^y  cos  X \ 


V/2-f2_,2 


\g-2ojri  cos  X-Wl^-^^—rf)"] 


[185] 


166         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 

two  simultaneous  differential  equations  in  c  and  ry  only. 
The  integration  of  these  gives  ^  and  tj  as  functions  of  t; 
then  by  substitution  in  [184]  ^  as  a  function  of  t  is  obtained 
and  tlie  problem  is  theoretically  solved.  The  mathematical 
difficulties,  however,  are  such  as  to  prevent  this  method' 
of  procedure,  and  leaving  aside  the  general  case  we  may 
obtain  an  approximate  solution  as  follows: 

183.  Approximate  solution. — Regard  j  and  j  and  their 

derivatives,  together  with  co,  as  small  quantities  whose  squares 
and  products  may  be  neglected.     Then  from 


c=.(i-^'^^* 


we  have   C  =  ^  ^^^^  equations  [185]  become 


[186] 


These  are  linear  equations  with  constant  coefficients  and 
can  be  integrated  by  quadratures.  The  following  method 
will  lead  to  interesting  results. 

Project  the  motion  on  the  plane 

which  is  tangent  to  the  sphere  of  constraint  at  its  lowest 
point  Ai.  Projecting  A^  and  Atj  on  this  plane,  giving  the 
axes  Ai$  and  Airj  (fig.  39),  then  the  differential  equations 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        167 
of  motion  of  Pi,  the  projected  pendulum,  are 


r'=2a;sm  >l'j'-rc 


7"= -2a;  sin  /l?'-p 


South 


Fig.  39. 


From  which 


[187] 


V >■  East 


^East 


[188] 


These  are  exact  equations,  and  integrating  we  have 
9 


[189] 


168 


ELEMENTS  OF  KINExMATICS  AND  MECHANICS 


Calling  r  and  0  the  polar  coordinates  of  the  projected 
pendulum 


^=rcos^        i^=r  sin  d 


and  [189]  become 


r^d'  +  wsin  X-r'^  =  C 


[190] 


Particular  case. — Suppose  the  pendulum  in  equilibrium 
in  the  vertical  and  set  it  in  motion  by  a  small  impulse. 

Since  r  and  d  are  both  zero  at  the  initial  instant,  then 
C  is  zero  and  the  second  equation  of  [190]  becomes 


whence 


d'  +  ujsm.  X  =  Q 


and  the  projected  pendulum  revolves  about  the  origin,  with 
constant  angular  velocity 

coi  =  (jj  sin  X 

in  the  negative  direction.    It  makes  a  complete  revolution 
in  the  time 


tJ^^ 


2iz 


2A^ 


Hence 


.     ,  -  •    1    (sidereal  time) 
a»i     oj  sin  /     sm  /  ^ 

T'st.  Louis  =  38^.44  (about) 


The  'pendulum  then  seems  to  oscillate  in  a  plane  which 
rotates  about  the  vertical  in  the  negative  direction  with  con- 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        169 

slant  angular  velocity  aji  =  a>  sin  L     The  time  of  a  complete 

24 

revolution  is  - — r  sidereal  hours. 
sin  / 

General  case. — Introducing  the  new  variable 


61=^6  + CO  sin  X-t  =  d  +  ojit 


equations  [190]  become 


r'^di'=C 


[191] 


The  variables  r  and  Ox  are  the  polar  coordinates  of  Pi  referred 
to  a  set  of  axes  Ai^i  and  Aitji  which  rotate  about  the  vertical 
in  the  negative  direction  with  constant  angular  velocity  coi. 

Neglecting  the  term  containing  coi^  and  calling  hi  a  new 
constant  equations  [191]  become 


r'^+rWi'^+f-r^  =  hi 
rWi'=C 


[192] 


These  may  be  written 


V^+f.r^  =  hy 
Areal  V=2C 


[193] 


Hence   referred    to    these    axes    the    projected    pendulum 


170         ELEMENTS  OF  KINEMATICS  AND  MECHANICS 


moves  under  the  action  of  a  force  y-r  directed  toward  ^li. 
The  path  is  then  an  eUipse  of  period 

T,=2r.J-  [195] 


The  projected  'pendulum  then  describes  an  ellipse  in  the 
horizontal  plane,  and  the  major  axis  of  this  ellipse  rotates 
about  the  vertical  in  the  negative  direction  with  angular  velocity 

lOl. 

Foucault's  case. — In  Foucault's  celebrated  experiment 
in  the  Pantheon  at  Paris,  the  pendulum  was  drawn  aside 
from  the  vertical  and  the  restraining  string  burned.  The 
initial  velocity  relative  to  the  axes  Ai$  and  Ait]  is  then 
zero,  and  hence  /  and  6'  are  both  initially  zero.  Since  r'  ==0 
initially  this  value  of  r  is  a  maximum  or  a  minimum  and 
is  therefore  a  semi-axis  of  the  ellipse.  Call  it  a.  Then  from 
the  second  equation  of  [190] 

(1) sin  k-a'^  =  (xJi-a^  =  C 

Then  from  the  second  equation  of  [192], 

r^Oi  =  (oia^ 

m 

or  Areal  F  =  2a>i-a2  [196] 

Theorem  of  Chevilliet. — Let  a  and  b  be  the  two  axes  of 
the  eUipse.    The  areal  velocity  is  then 

Areal  V=  ^n 
i  1 


CONSTRAINED  MOTION  ON  EARTH'S  SURFACE        171 

Hence  2(i)ia?  =-ffr 

-t  1 

Whence  —=  I,   ^  =-^  =  a)sm.  X-y}- 

a       27:       T  ^g 

In  the  experunent  in  the  Pantheon 

and  hence  the  eUipse  was  so  flat  as  to  seem  a  line. 

At  St.  Louis,  with  the  same  values  of  /  and  a  we  should 
have 

b     16«.43        1 


a    38''.44     8423 


INDEX. 


ARTICLE 

Absolute  acceleration 45, 100 

Absolute  axes 96, 106 

Absolute  coordinates 96 

Absolute  displacement 45, 96, 106 

Absolute  motion 96 

Acceleration  in  curvilinear  motiou 61 

Acceleration  in  rectilinear  motion 30 

Amplitude 41 

Analytic  expression  of  components  of  a  vector 8 

Analytic  expression  of  force  between  two  free  particles 140 

Analytic  expression  of  geometric  difference 6 

Analytic  expression  of  geometric  sum 4 

Analytic  expression  of  geometric  sum  referred  to  axes 10 

Apparent  motion 48 

Areal  velocity 72 

Argument 41 

Attraction  toward  a  fixed  centre 154 

t'entimetre-Gram-Second  system  of  units 133 

Central  forces 141 

Change  of  units 23 

Composition  of  forces 122 

Composition  of  harmonic  angular  motions 81 

Composition  of  harmonic  rectilinear  motions 51 

Composition  of  motions  along  same  path 66 

Constant  field  of  force Ill 

Constrained  particle 164 

Convective  acceleration 49,  100 

173 


174  INDEX. 

ABTICUI 

Convective  motion 49, 100 

Convective  velocity 49, 100 

Curve  of  constraint 173 

Decomposition  of  a  vector 7 

Decomposition  of  forces 123 

Derived  units 131 

Differential  equations  of  motion 144 

Displacement 55 

Elongation 41 

English  system  of  units 135 

Equations  of  absolute  and  relative  motion 97 

Equilibrium  of  a  constrained  particle 172 

Equilibrium  of  a  free  particle 124 

EquipoUence 2 

Field  of  force 109 

Finite  and  differential  equations  of  motion 144 

First  principle  of  mechanics 107 

Fixed  and  moving  axes 96 

Force 120 

Forces  of  constraint 168 

Force  producing  extinction 151 

Force  producing  harmonic  motion 149 

Foucault's  pendulum 182 

Fundamental  units 131 

Geometric  derivative 11 

Geometric  difference 5 

Geometric  sum 3 

Gravitation. 142 

Harmonic  angular  motion 79 

Harmonic  motion 40 

Hodograph 61 

Homogeneity 74 

Homogeneous  equations 22 

Isolated  particle 100 

Knots. 28 


INDEX.  175 

ARTICLE 

Mass 119 

Material  point 101 

Metre-Kilogram-Second  system  of  units 134 

Moving  axes 96 

Nautical  miles 28 

Oscillations  of  smail  amplitude 179 

Particle 101 

Principles  of  mechanics , 105, 

Principle  of  the  equality  of  action  and  reaction 139 

Problem  of  mechanics 103 

Problems  of  relative  motion 157 

Projection  of  acceleration  on  binormal 64 

Projection  of  acceleration  on  tangent  and  normal 63 

Projection  of  geometric  derivative 9 

Projection  of  geometric  derivative  on  vector 13 

Projection  of  velocity  on  displacement 58 

Resistance  of  particle  on  field 115 

Relation  between  linear  and  angular  velocity 69 

Relation  between  linear  and  angular  acceleration 71 

Relative  acceleration 47, 96 

Relative  angular  motion 80 

Relative  axes 96 

Relative  coordinates 96 

Relative  displacement 45, 96 

Relative  force 156 

Relative  motion 45, 06, 96 

Relative  velocity 47, 100 

Resistance  producing  extinction 151 

Second  principle  of  mechanics 113 

Simple  pendulum 178 

Simultaneous  motions 50 

Smooth  constraints 167 

Superposition  of  fields  of  force 112 

Systems  of  units 132 

Tangential  and  normal  force 127 

Tension  in  string  of  simple  pendulum 187 


170  INDEX. 

ARTICLE 

Theorem  of  Chevilliet 183 

Third  principle  of  mechanics 116 

Turning  acceleration 100 

Uniform  field  of  force 110 

Unit  of  force 137 

Units  of  astronomy 136 

Units  of  kinematics j '  129 

Units  of  mechanics 130 

Universal  gravitation 142 

Variable  constraints 166 

Variation  of  g  with  latitude 176 

Vectors 1 

Velocity 56 

Velocity  in  oscillations  of  any  amplitude. 180 


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1 


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2 


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3 


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4 

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Arches. ..Svo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete  ►Plain  and  Reinforced.    (7n  press.) 

•  Traiitwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep.  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  >;  00 

Sheep.  50 

Law  of  Contracts 8vo,  3  00 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  U^e  and  Adjustment  of  Engineerinsr  Instruments. 

i6mo,  morocco,  i  25 

•  Wheeler's  Elementary  Course  of  Civil  Engineering 8vo,  4  00 

Wilson's  Topographic  Surveying 8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boner's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  00 

•  Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses.  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  so 

Do  Bois's  Mechanics  of  Engineering.     VoL  II Small  4to,    10  00 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  00 

Fowler's  Coffer-dam  Process  tor  Piers 8vo,  2  50 

Ordinary  Foundations 8vo,  3  50 

Oreene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo,  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  00 

Johnson,  Bryan,  and  Tumeaure's  Theory  and  Practice  in  the  Designing  of 

Modern   Framed   Structures Smali  4to,    10  00 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I. — Stresses  in  Simple  Trusses 8vo,  2  50 

Part  n. — Graphic  Statics 8vo,  2  50 

Part  nT  — Bridge  Design.    4th  Edition,  Rewritten 8vo,  2  50 

Part  IV. — Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,    10  00 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers. . .  i6mo.  morocco^  3  00 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.Svo,  2  00 
Wright's  Designing  of  Draw-spans: 

Part  L  — Plate-girder  Draws 8vo»  2  50 

Part  n. — Riveted-truss  and  Pin-connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume Svo,  3  50 

6 


HYDRAULICS. 
Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from  an 

Orifice.     (Trautwine.) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics 8vo,  5  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6m0(  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power. lamo,  3  00 

Folwell's  Water-supply  Engineering Svo,  4  00 

Frizell's  Water-power Svo,  5  00 

Fuertes's  Water  and  Public  Health i3mo,  i   50 

Water-filtration  Works lamo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) Svo,  400 

Hazen's  Filtration  of  Public  Water-supply Svo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Water-works Svo,  2  50 

Herschel's  1 15  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits Svo,  2  00 

Mason's   Water-supply.     (Considered   Principally  from   a   Sanitary   Stand- 
point)    3d  Edition,  Rewritten Svo,  4  00 

Merriman's  Treatise  on  Hydraulics.     9th  Edition,  Rewritten Svo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics Svo,  4  00 

Schuyler's  Reservoirs  for  Irrigation,  Water-power,  and  Domestic  Water- 
supply  Large  Svo,  5  00 

**  Thomas  and  Watt's  Improvement  of  Riyers.     (Post.,  44  c.  additional),  4to,  6  00 

Tumeaure  and  Russell's  Public  Water-supplies Svo,  5  00 

Wegmann's  Desitrn  and  Construction  of  Dams 4to,  5  00 

Water-supply  of  the  City  of  New  York  from  165S  to  1S95 4to,  10  00 

Weisbach's  Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) Svo,  5  00 

Wilson's  Manual  of  Irrigation  Engineering Small  Svo«  4  00 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 

Wood's  Turbines Svo,  a  50 

Elements  of  Analytical  Mechanics Svo,  3  00 

MATERIALS  OP  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction Svo,  5  00 

Roads  and  Pavements Svo,  5  00 

Black's  United  States  Public  Works '. Oblong  4to,  5  00 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edi- 
tion, Rewritten Svo,  7  50 

Byrne's  Highway  Construction Svo,  5  00 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  00 

Church's  Mechanics  of  Engineering Svo,  6  00 

Du  Bois's  Mechanics  of  Engineering.     VoL  I Small  4to,  7  50 

Johnson's  Materials  of  Construction Large  Svo,  6  00 

Fowler's  Ordinary  Foundations Svo,  3  50 

K.eep's  Cast  Iron Svo,  2  50 

Lanza's  Applied  Mechanics Svo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     a  vols. Svo,  750 

Merrill's  Stones  for  Building  and  Decoration Svo,  5  00 

Merriman's  Text-book  on  the  Mechanics  of  Materials Svo,  4  00 

Strength  of  Materials : i2mo,  i  00 

Metcalf's  SteeL     A  Manual  for  Steel-users i2mo,  2  00 

Patton's  Practical  Treatise  on  Foundations Svo,  5  00 

Richey's  Handbook  for  Building  Superintendents  of  Construction,     (/n  jrresa.) 

Rockwell's  Roads  and  Pavements  in  France lamo,  i  25 

7 


Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Text-book  on  Roads  and  Pavements i2mo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced.     (In 

press.) 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  00 

Part  I. — Non-metallic  Materials  of  Engineering  and.  Metallurgy 8vo,  2  00 

Part  II. — Iron  and  Steel 8vo,  3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Other  AUoys  and  their 

Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  00 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

Waddell's  De  Pontibus.     (A  Pocket-book  for  Bridge  Engineers.).  .  i6mo,  mor.,  3  00 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  (DeV.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings :    Corrosion  and  Electrolysis  of  Iron  and 

SteeL  • 8vo,  4  00 

RAILWAY  ENGINEERING. 

Andrews's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  s  00 

Brooks's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butts's  Civil  Engineer's  Field-book i6mv,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  .norocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  1  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.     i6mo,  morocco,  5  00 

Dredges  History  of  the  Petmsylvania  Railroad:  (1879) Paper,  5  00 

*  Drinker's  TunneUng,  Explosive  Compounds,  and  Rock  Drills,  4to,half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards ' Cardboard,  35 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide. . .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  I  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  -oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  -oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  ot  Calculating  the  Cubic  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo, morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction.     2d  Edition ^ Rewritten i6mo,  morocco,  5  00 

Wellington's  Economic  Theory'of  the  Location  of  Railways SmaU  8vo,  s  00 

DRAWING. 

Barr's  Kinematics  of  Machinery. 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

*  "       Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  ot  Drawing 8vo,  paper,  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to.  2  so 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  00 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I. — Kinematics  of  Machinery 8vo,  i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts Svo,  3  00 

MacCord's  Elements  of  Descriptive  Geometry. 8vo,  3  00 

Kinematics;  or.  Practical  Mechanism Svo,  5  00 

Mechanical  Drawing 4to,  4*00 

Velocity  Diagrams Svo,  i  50 

Mahan's  Descriptive  Geometry  and  Stone-cutting. Svo,  i  50 

Industrial  Drawing.     (Thompson.) Svo,  3  50 

Ifoyer's  Descriptive  Geometry,     (/n  press.) 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.  .8vo,  3  00 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Schwamb.  and  Merrill's  Elements  of  Mechanism Svo,  3  00 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo,  2  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing . .  i2mo,  i  00 

Drafting  Instruments  and  Operations i2mo,  i  25 

Manual  of  Elementary  Projection  Drawing i2mo,  1  50 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  i  00 

Plane  Problems  in  Elementary  Geometry ; . . . .  i2mo,  i  25 

Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective Svo,  3  50 

General  Problems  of  Shades  and  Shadows Svo,  3  00 

Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry Svo,  2  50 

Weisbach's  Kinematics  and  the  Power  of  Transmission.     (Hermann  and 

Klein.) Svo,  5  00 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2nio,  2  00 

Wilson's  (H.  M.)  Topographic  Surveying Svo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective Svo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering Svo,  i  00 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  Svo,  3  00 

ELECTRICITY  AND   PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  Svo,  3  00 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measiu-ements . . . .  i2mo,  i  00 

Benjamin's  History  of  Electricity Svo,  3  00 

Voltaic  Cell Svo,  3  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).  .Svo,  3  00 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo,  3  00 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  s  00 
Dolezalek's    Theory   of    the    Lead    Accumulator    (Storage    Battery).     (Von 

Ende.) i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) Svo,  4  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Gilbert's  De  Magnete.     (Mottelay.) Svo,  2  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  00 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50  . 

Holman's  Precision  of  Measurements Svo,  2  00 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests Large  Svo,  75 

Kinzbrunner's  Testing  of  Continuous-Current  Machines Svo,  2  00 

Landauer's Spectrum  Analysis.    (Tingle.)..... Svo.  3  00 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.  )iamo,  3  00 

LOb'sElectrolysisand  £lectro8yntbesis  of  Organic  Compounds.  (Lorenz.)  lamo,  i  00 

9 


•  Lyons'B Treatise  on ElectromagiieticPhenomeiuu     Vols.  I.  and  IL  8to,  each,  600 

•  Hichie.     Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  00 

ITiaudet'B  Elementary  Treatise  on  Electric  Batteries.     (Fishoack.) i2mo,  250 

•  Rosenberg^s  Electrical  Engineering.   (Haldane  Gee — Kinzbninner.) 8vo,  1  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  L 8vo,  250 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

•  Tillman's  Elementary  Lessons  in  Heat 8vo.  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo.  2  00 

Ulke'ji  Modern  Electrolytic  Copper  Refining 8vo,  3  00 

LAW. 

•  Davis's  Elements  of  Law 8vo,  2  50 

•  Treatise  on  the  Military  Law  ot  United  States 8vo,  7  00 

•  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  1  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep.  6  50 
Law  of  Operations  PreUminary  to  Construction  in  Engineering  and  Archi- 
tecture     8vo,  5  00 

Sheep,  S  50 

Law  of  Contracts 8vo,  3  00 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

MANUFACTURES. 

Bemadou's  Smokeless  Powder — Kitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Holland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement. i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  00 

Eissler's  Modem  High  Explosives 8vo,  4  00 

ESront's  Enzymes  and  their  Applications.     (Prescott.) 8vo  3  00 

Fitzgerald's  Boston  Machinist i8mo,  i  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  00 

Keep's  Cast  Iron ., 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

ControL     (In  preparation.) 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  SteeL    A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration    of  Workshops, 

Public  and  Private 8vo,  5  00 

Meyer's  Modem  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories. i6mo,  morocco,  i  50 

•  Reisig's  Guide  to  Piece-dyeing 8vo,    25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oc 

Handbook  for  Sugar  Manufacturers  and  their  Chemists..  .i6mo  morocco,  2  00 
Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced,     (/n 
press.) 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion   8vo,  5  00 

•  Walke's  Lectures  on  Explosives 8vo,  4  00 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


WolfE's  Wmdmill  as  a  Prime  Mover 8vo,  3  00 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

Wood's  Rustless  Coatings:  Cocrosion  and  Electrolysis  of  Iron  and  Steel. .  .8to,  4  00 

MATHEMATICS. 

Baker's  Elliptic  Functions 8to,  t  50 

•  Bass's  Elements  of  Differential  Calculus i2mo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo,  i  00 

Compton's  Manual  of  Losarithmic  Computations i3mo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

•  Dickson's  College  Algebra Large  1 2mo,  i  50 

•  Introduction  to  the  Theory  of  Algebraic  Equations   Large  i2mo,  i  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Halsted's  Elements  of  Geometry 8vo,  i  75 

Elementary  Synthetic  Geometry 8vo,  i  50 

Rational  Geometry i2mo, 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size,  .paper,  15 

100  copies  for  5  00 

*  Mounted  on  heavy  cardboard,  8X10  inches,  25 

10  copies  for  2  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .    Small  8vo,  3  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus.  .Small  8vo,  i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,  i  00 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,  3  50 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  .  i2nio,  i   50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2n30,  2  00 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,  3  00 

Trigonometry  and  Tables  published  separately Each,  2  00 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables Svo,  i  00 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  00 

Merriman's  Method  of  Least  Squares 8vo,  2  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus .  Sm.,  8vo,  3  00 

Differential  and  Integral  Calculus.     2  vols,  in  one Small  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  00 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,  i  00 

MECHAmCAL   ENGINEERmO. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr'a  Kinematics  of  Machinery 8vo,  2  so 

•  Bartlett't  Mechanical  Drawing 8vo,  3  00 

•  "                 •*               "        Abridged  Ed 8vo.  i  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  00 

Carpenter's  Experimental  Engineering 8vo,  6  00 

Heating  and  Ventilating  Buildings 8vo,  4  00 

Gary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL      (/n  prep- 
aration.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  00 

Coolidge's  Manual  of  Drawing 8vo,    paper,  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  so 

11 


Cromwell's  Treatise  on  Toothed  Gearing lamo  i  50 

Treatise  on  Belts  and  Pulleys lamo,  i  50 

Dnrley's  Kinematics  of  Machines 8vo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power lamo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analjrsis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo.  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  00 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part   I. — Kinematics  of  Machinery 8to,  i  50 

Part  n. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kent's  Mechanical  Engineer's  Pocket-book i6mo,  morocco,  5  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Leonard's  Machine  Shops,  Tools,  and  Methods.     (In  pre»a.) 

MacCord's  Kinematics;  or,  Practical  Mechanism. Svo.  5  00 

Mechanical  Drawing 4to.  4  00 

Velocity  Diagrams 8vo,  i  50 

Hahan's  Industrial  Drawing.    (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels Svo.  3  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design . .  8vo.  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanfsm 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Thurston's  Treatise  on   Friction  and    Lost  Work  in   Machinery  and  Mill 

Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 870,  7  50 

Weisbach's  Kinematics  and  the  Power  of  Transmission.      Herrmann — 

Klein.) 8vo,  5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  500 

Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vo,  5  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Turbines , . . . .  8vo,  2  50 

MATERIALS  OF  ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Johnson's  Materials  of  Construction Large  8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Texx-book  on  the  Mechanics  of  Materials 8vo,  4  00 

Strength  of  Materials   i2mo,  i  00 

MetcalTs  SteeL     A  Manual  for  Steel-users x2mo  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering 3  vols,,  Svo.  8  00 

Part   IL — Iron  and  Steel Svo,  3  50 

Put  ni. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo  2  50 

Taxt-book  of  the  Materials  of  Construction 8vo,  5  00 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  00 

Wood's  (M.  P.)  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel. 

8vo,    4  00 


STEAM-ENGINES  AND  BOILERS. 

Camot's  ReflectionB  on  the  Motive  Power  of  Hest.     (Thurrton.) lamo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. .  i6mOi  mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Goss's  Locomotive  SpexlJS 8vo,  2  00 

Hemenway's  Indicator  Practce  and  Steam-engine  Economy ismo,  2  00 

Hotton's  Mecluuucal  Engineering  of  Power  Plants 8vo,  5  00 

Heat  and  Heat-engines 8vo,  5  co 

Kent's  Steam-boiler  Economy 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  00 

Meyer's  Modem  Locomotive  Construction 4to.  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator lamo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-enginek 8vo,  5  00 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers Svo.  4  00 

Fray's  Twenty  Years  with  the  Indicator Large  Svo,  2  50 

Pupln's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Satiirated  Vapors. 

(Osterberg.) lamo,  i  25 

Reagan's  Locomotives :  Simple,  Compound,  and  Electric/. lamo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management lamo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice lamo,  2  50 

Snow's  Steam-boiler  Practice Svo,  3  00 

Spangler's  Valve-gears Svo,  2  50 

Notes  on  Thermodynamics lamo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Handy  Tables Svo,  i  50 

Manual  of  the  Steam-engine 2  vols.  Svo,  10  00 

Part  L — History,  Structuce,  and  Theory Svo,  6  00 

Part  n. — Design,  Construction,  and  Operation Svo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake Svo,  5  00 

Stationary  Steam-engines Svo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice lamo,  i  50 

Manualof  Steam-boilers,  Their  Designs,  Construction,  and  Operation. Svo,  5  00 

Wsisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) Svo,  5  00 

Whitham's  Steam-engine  Dssign Svo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flatber.) i6mo,  2  50 

Wood's  Thermodynamics  Heat  Motors,  and  Refrigerating  Machines ....  Svo,  4  00 


MECHANICS    AND  MACHINERY. 

Ban's  Kinematics  of  Machinery Svo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  a  50 

Church's  Mechanics  of  Engineermg Svo,  6  00 

IS 


Church's  Notes  and  Examples  in  Mechanics 8vo,  2  00 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  the  Use  of  Colleges  and 

Schools i2mo,  I  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to   half  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

VoL     I. — Kinematics 8vo,  3  50 

Vol.    n.— Statics    8vo.  4  00 

VoL  in.— Kinetics 8vo.  3  50 

Mechanics  of  Engineering.     Vol.   I Small   4to.  7  50 

VoL  n. Small  4to,    10  00 

Durley's  Kinematics  of  Machines  8vo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  i  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12010,  3  00 

Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hall's  Car  Lubrication i2mo,  i  00 

Holly's  Art  of  Saw  Filing i8mo,  75 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods Svo,  2  00 

Jones's  Machine  Design: 

Part  1. — Kinematics  of  Machinery Svo,  1  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts Svo,  3  00 

Kerr's  Power  and  Power  Transmission Svo.  2  00 

Lanza's  Applied  Mechanics Svo,  7  so 

Leonard  s  Machine  Shops,  Tools,  and  Methods,    (/n  press.) 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  00 

Velocity  Diagrams Svo,  i  50 

Maurer's  Technical  Mechanics. Svo,  4  00 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo .  4  00 

Elements  of  Mechanics i2mo,  i  00 

*  Michie's  Elements  of  Analytical  Mechanics Svo  4  00 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric X2mo,-  2  50 

Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. . Svo,  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  I Svo.  250 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management i2mo.  2  00 

Smith's  Press-working  of  Metals Svo,  3  00 

Materials  of  Machines lamo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill 

Work Svo,  3  00 

AnimalasaMachine  and  Prime  Motor,  and  tile  Laws  of  Energetics. i2mo,  i  00 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Weisbach's    Kinematics    and    the  Power  of    Transmission.     (Herrmann — 

Klein.) ^ Svo,  5  00 

Machinery  of  Transmission  and  Governors.    (Herrmann — Klein.). Svo,  s  00 

Wood's  Elements  of  Analytical  Mechanics Svo,  3  00 

Principles  of  Elementary  Mechanics lamo  i  25 

Turbines , Svo,  2  50 

The  World's  Columbian  Exposition  of  xSga 4to,  i  00 

14 

THE  LIBRARY 

UNIVERSITY  OF  CALIFORNIA 

LOS  ANGELES 


METALLURGY. 
Bgleston's  Metallurgy  of  SilTer.  Gold,  and  Mercury: 

VoL   I.— Silver 8vo,  7  So 

VoL   II. — Gold  and  Mercury 8vo,  7  So 

**  Iles's  Lead-smelting.    (Postage  9  cents  additional) lamo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Cbatelier's  High-temperature  Measurements.  (Boudouard — Burgess.).  lamo,  3  00 

Metcalf's  SteeL    A  Manual  for  Steel-users i2mo,  2  00 

Smith's  Materials  of  Machines izmo,  i  00 

Thurston's  Materials  of  Engineering.    In  Three  Parts 8vo,  8  00 

Part  n. — Iron  and  Steel 8vo,  3  50 

Partni. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and   their 

Constituents 8vo,  2  50 

Dike's  Modem  Electrolytic  Copper  Refining 8vo,  3  00 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.     Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo.  3  00 

Map  of  Southwest  Virginia. Pocket-book  form,  2  00 

Brush's  Manual  of  Determinative  Mineralogy.    (Penfield.) Svo,  4  00 

Chester's  Catalogue  of  Minerals Svo,  paper,  i  00 

Cloth,  I  25 

Dictionary  of  the  Names  of  Minerals Svo,  3  50 

Dana's  System  of  Mineralogy Large  Svo,  half  leather,    12  50 

First  Appendix  to  Dana's  New  "System  of  Mineralogy." Large  Svo,  i  00 

Text-book  of  Mineralogy Svo,  4  00 

Minerals  and  How  to  Study  Them. lamo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  Svo,  i  00 

Manual  of  Mineralogy  and  Petrography z2mo,  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

Eakle's  Mineral  Tables Svo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.     (Smith.)  Small  Svo,  2  00 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses. Svo,  4  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper,  o  50 
Rosenbusch's   Microscopical  Physiography  of   the   Rock-making  Minerals. 

(Iddings.) Svo,  5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Docks Svo,  2  00 

Williams's  Manual  of  Lithology Svo,  3  00 

MmilTG. 

Beard's  Ventilation  of  Mines lamo,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form,  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills. 

4to,  half  morocco,    25  00 

Eissler's  Modem  High  Explosives Svo,  4  00 

Fowler's  Sewage  Works  Analyses lamo,  2  00 

Goodyear 's  Coal-mines  of  the  Western  Coast  of  the  United  States zamo,  2  50 

Ihlseng's  Manual  of  Mining Svo,  4  00 

**  Iles's  Lead-smelting.     (Postage  gc.  additionaL) lamo,  a  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe Svo,  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores Svo,  2  00 

*  Walke's  Lectures  on  Explosives Svo,  4  00 

Wilson's  Cyanide  Processes xamo,  i  50 

Chlorination  Process xamo.  i  50 

15 


Wilson's  Hydraulic  and  Placer  Mining i2mo,  2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  00 

Water-supply  Engineering 8vo,  4  00 

Fuertes's  Water  and  Public  Health. i2mo,  1  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  00 

Goodrich's  Economical  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Pubhc  Water-supplies 8vo,  3  00 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.     (Considered    Principally    from    a    Sanitary    Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  00 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Merriman's  Elements  of  S«>Bitary  Engineering 8vo,  2  00 

Ogden's  Sewer  Design i2mo,  2  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Reference 

to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  i  5c 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  00 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  00 

Richards   and  Woodman's  Air,  Water,  and  Food   from  a  Sanitary  Stand- 
point  8vo,  2  00 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewagt 8vo,  3  50 

Turneaure  and  Russell's  Pubhc  Water-suppUes 8vo,  5  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  1  00 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  and  Military  Hygiene i6mo,  1  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff.) i2mo,  2  50 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart.  1  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Poljrtechnic  Institute,  1824- 1894.  Small  8vo,  3  00 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  1  00 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  1  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 

HospitaL i2mo,  i  25 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Grammar  of  the  Hebrew  Language 8vo,  3  00 

Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

16 


UNIVERSITY  OF  CALIFORNIA,  LOS  ANGELES 

THE  UNIVERSITY  LIBRARY 

This  book  is  DUE  on  the  last  date  stamped  below 

AP*^  2  6  1951. 
APP2-1957 


OCT  19 1961 
FEB  1  2  ISaS 


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Sciences 
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